Every key fact as cards, with a self-test in each section.
Section 01
Statistics, Populations and Samples
Important
4 cards
Descriptive and inferential statistics
Descriptive statistics organize and summarize the data actually observed. Inferential statistics use sample data, probability, and sampling distributions to draw conclusions about a population while expressing uncertainty.
Inference rests on the difference between a population, the whole group of interest, and a sample drawn from it. Values that describe a population are parameters, values calculated from a sample are statistics, and the chance gap between them is sampling error.
Descriptive statistics
Inferential statistics
Purpose
Summarize the scores actually collected
Use a sample to reach conclusions about a population
Main tools
Tables, graphs, and measures of center, spread, and shape
Probability, sampling distributions, standard errors, confidence intervals, and p-values
Scope
No claim beyond the observed data
Conclusions concern the wider population
Example
Mean anxiety score of 40 participants
Whether a therapy reduces anxiety in the wider patient population
Explain
Descriptive statistics stop at the data in hand: the average anxiety score of 40 participants is simply a fact about those 40 people. Inferential statistics take the next step and ask what the data suggest about everyone like them, for example whether a therapy would help all patients with anxiety. Because a sample never matches the population exactly, inference always comes with a measure of uncertainty, such as a standard error or a p-value.
Populations and samples
A population is the complete set of individuals, objects, or measurements of interest, and a sample is a subset drawn from it. A numerical characteristic of a population is a parameter, and the corresponding value calculated from a sample is a statistic.
Population
Sample
Summary value
Parameter
Statistic
Size
N
n
Mean
μ
M or X̄
Standard deviation
σ
s or SD
Variance
σ²
s²
By convention, Greek letters denote parameters and Roman letters denote statistics.
Explain
The population is set by the research question, not by its size: all first-year psychology students in a country, or all patients at one clinic. Researchers rarely measure a whole population, so they measure a sample and use its statistics to estimate the population's parameters. A simple way to keep the terms apart is that parameters go with populations (both begin with p) and statistics go with samples (both begin with s).
Sampling error
A sample statistic rarely equals its population parameter exactly. The difference is sampling error, the chance variation produced because only part of the population was observed.
PopulationHas a fixed parameter
SampleContains only part of the population
StatisticVaries from sample to sample
Sampling errorChance difference between statistic and parameter
Explain
Sampling error is not a mistake by the researcher; it is the natural difference that comes from looking at only part of the group. If five classes from the same college each report their average height, the five averages will differ slightly, although they all come from the same population. Larger samples reduce this error, which is why the standard error shrinks as the sample grows.
Check yourself
What is the difference between descriptive and inferential statistics?
Descriptive statistics summarize observed data, whereas inferential statistics use sample data to draw conclusions about a population.
What is a parameter?
A numerical characteristic of a population.
What is a statistic?
A numerical value calculated from a sample.
Which symbols are used for population and sample size?
Population size is N and sample size is n.
What is sampling error?
Chance variation between a sample statistic and its population parameter because only part of the population was observed.
Section 02
Scales of Measurement and Types of Variables
Important
5 cards
Scales of measurement
A variable is a characteristic that can take different values across observations, and data are its recorded values. Stevens described four scales of measurement. They are cumulative, so each keeps the properties of the scale below it and adds one more.
Properties
Examples
Common summaries or procedures
Nominal
Categories with no inherent order
Blood group; diagnostic categories
Frequencies; percentages; mode; chi-square
Ordinal
Ordered categories; intervals not assumed equal
Ranks; ordered ratings; socioeconomic class
Median; percentiles; rank-based statistics
Interval
Equal intervals but no true zero
Celsius temperature; many standardized scores
Mean; SD; correlation; t-tests
Ratio
Equal intervals and a meaningful zero
Height; weight; reaction time; Kelvin temperature
All arithmetic operations; CV
Variables can also be grouped by type: categorical, or quantitative, which is either discrete (counted) or continuous (measured).
Explain
The four scales can be remembered by the order NOIR. Numbers on a cricketer's shirt are nominal: they only name. Finishing positions in a race are ordinal: first is ahead of second, but not by a fixed distance. Temperature in Celsius is interval: the steps are equal, but 0 °C does not mean no temperature. Weight is ratio: 0 kg means no weight, so 20 kg is truly twice 10 kg.
Statistics permitted by scale
A statistic suited to a lower scale remains usable at a higher one. The median needs at least ordinal data, and the mean and standard deviation need at least interval data.
Mode
Median
Mean and SD
Nominal
Yes
No
No
Ordinal
Yes
Yes
No
Interval
Yes
Yes
Yes
Ratio
Yes
Yes
Yes
Explain
Each statistic needs the information that its calculation uses. The mode only counts how often each category appears, so it works even for names. The median needs an order to find the middle, and the mean adds up the values, which only makes sense when the distances between them are equal. For this reason the average of category codes, such as 1 for blood group A and 2 for blood group B, has no meaning.
Common confusion: interval and ratio scales
IncorrectEqual intervals alone make a variable ratio scale.
CorrectA ratio scale also requires a meaningful zero that represents the absence of the measured quantity. Celsius temperature is interval, whereas height is ratio. A height of 180 cm is twice 90 cm, but 40 °C is not twice as hot as 20 °C.
Types of variables
A categorical variable places observations into categories. A quantitative variable records a numerical amount and is either discrete or continuous.
Countable values with no possible values in between
Number of errors; children; admissions
Quantitative: continuous
Can in principle take any value within an interval
Height; weight; time
This distinction is separate from the four scales of measurement. Continuous measurements are still recorded to finite precision, so a reaction time recorded as 412 ms stands for the interval from 411.5 to 412.5 ms.
Explain
A discrete variable is counted, so it moves in whole steps: a family can have two or three children but not 2.5. A continuous variable is measured, so between any two values there is always another, such as 61.5 kg between 61 and 62 kg. Because every measuring instrument stops at some precision, a recorded continuous score stands for a small interval, its exact limits, which run half a unit above and below it.
Check yourself
What is the order of the four measurement scales from least to most informative?
Nominal → ordinal → interval → ratio.
What minimum scale is required for the median?
Ordinal.
What minimum scale is required for the mean and standard deviation?
Interval.
What separates an interval scale from a ratio scale?
A ratio scale has a meaningful zero, so ratios are meaningful.
How does a discrete quantitative variable differ from a continuous one?
A discrete variable has countable values with no possible values in between, whereas a continuous variable can in principle take any value within an interval.
Section 03
Frequency Distributions and Graphs
Important
4 cards
Frequency distributions
A frequency distribution shows how often each value or interval occurs. When a variable has many possible values, scores are grouped into class intervals of equal width, and the table may add the relative frequency and the cumulative frequency of each interval.
Cumulative frequencies (cf) are the basis for medians, percentiles, and percentile ranks.
The same data can be shown as a bar chart, pie chart, histogram, frequency polygon, ogive or box plot, each suited to a different kind of data.
Explain
Grouping scores into class intervals turns a long list of numbers into a picture of where scores bunch together. Relative frequency shows each interval's share of the whole, which makes groups of different sizes comparable. Cumulative frequency adds the counts up as it goes, so it shows how many scores lie below any point, which is exactly what is needed to find the median or a percentile.
Choosing a display
Best suited to
What it shows
Bar chart
Categorical or discrete data
Separate bars for distinct categories
Pie chart
Categorical data
Each category's share of the whole
Histogram
Grouped continuous data
Adjacent bars over numerical class intervals
Frequency polygon
Grouped continuous data
Interval midpoints joined by straight lines; useful for overlaying two distributions
Ogive
Cumulative frequencies
Rising cumulative curve from which the median and percentiles can be read
Box plot
Quantitative data
Median, quartiles, spread, and potential outliers
A scatterplot serves a different purpose, since it shows the relationship between two quantitative variables.
Explain
The choice follows the kind of data. Categories such as diagnoses suit a bar chart or a pie chart, while scores on a continuous scale such as test marks suit a histogram or a frequency polygon. Two frequency polygons can be drawn on the same axes to compare two groups, which overlapping histograms would make hard to read. An ogive answers questions such as how many students scored below 40, and a box plot summarises the middle, the spread and any unusual scores in one small figure.
Common confusion: bar charts and histograms
IncorrectBar charts and histograms differ only in visual style.
CorrectBar charts represent distinct categories, so the bars are separated and their order may be arbitrary. Histograms use a continuous numerical scale, so the bars normally touch and their order is fixed by the class intervals.
Check yourself
What does a frequency distribution show?
How often each value or interval occurs.
What is cumulative frequency?
The number of scores up to the top of a given interval.
Which display uses adjacent bars over numerical class intervals?
A histogram.
Which display is based on cumulative frequencies?
An ogive.
Which display shows median, quartiles, spread, and potential outliers?
A box plot.
Section 04
Measures of Central Tendency
Very important
7 cards
Mean, median and mode
Measures of central tendency identify a value that represents the center or typical score. The choice among mean, median, and mode depends on the measurement scale, the shape of the distribution, and the presence of extreme scores.
Lowest scale
Effect of extreme scores
Preferred when
Mean
Interval
Pulled toward them
Distribution is roughly symmetric, or further analysis such as SD, t-tests, or correlation will follow
Median
Ordinal
Largely unaffected
Distribution is skewed or has outliers
Mode
Nominal
Unaffected
Data are categorical, or the most typical single value is needed
The section also shows how to find the median and the mode from a grouped frequency table.
The arithmetic mean
The mean is the sum of all scores divided by the number of scores. Because it uses every score, it is mathematically precise but sensitive to extreme values.
Population: μ = ΣX / N
Sample: M = ΣX / n
Σ
sum of
X
each score
N
population size
n
sample size
Balance pointThe distribution balances at the mean
Σ(X − M) = 0Deviations from the mean sum to zero
Least squaresThe sum of squared deviations is smallest around the mean
Explain
The balance-point idea explains the mean's behaviour. Each score pulls on the mean in proportion to its distance from it, so the pulls above and below cancel out exactly, which is why the deviations sum to zero. A single far-away score pulls hard, so it moves the balance point a long way. The mean is also the value that makes the squared deviations as small as possible, which ties it to the variance and standard deviation.
The median
The median is the middle score after the values are ordered. It is the 50th percentile and Q2, and it depends on the position of extreme scores rather than their exact values.
Odd n: position (n + 1) / 2
Even n: mean of positions n/2 and n/2 + 1
1
Order the scores3, 5, 7, 9, 10, 12
2
Locate the middle positionsWith n = 6, the 3rd and 4th scores
3
Average them(7 + 9) / 2 = 8
Explain
The median only cares about which score is in the middle, not how far the other scores lie from it. That is why reports on household income use the median: a handful of very rich households push the mean far above what a typical family earns, but they hardly move the median. To find it, the scores must be put in order first; taking the middle of an unordered list is the commonest slip.
Mean and median with an extreme score
The scores 4, 5, 5, 6, and 30 have a mean of 10, although four of the five scores are 6 or lower. The median is 5 and would remain 5 even if 30 became 300.
Explain
The picture shows the difference between the two measures at a glance. Four scores sit close together around 5, and one sits far away at 30. The median stays with the cluster, while the mean is dragged toward the lone extreme score and ends at 10, a value that no score is near.
The mode
The mode is the most frequently occurring score. It is the only measure of central tendency appropriate for nominal data, although it can be used at any scale.
UnimodalOne peak
BimodalTwo peaks, which often indicate two distinct subgroups
MultimodalMore than two peaks
No modeEvery score occurs equally often
For continuous data the mode may be uninformative because exact scores rarely repeat. It is also the least stable of the three measures across samples.
Explain
The mode is the natural summary for categories: the most common diagnosis in a clinic or the most popular choice in a survey. A bimodal distribution often signals two groups mixed together, such as the heights of men and women combined. With continuous measurements, such as reaction times recorded to the millisecond, almost no two scores are exactly equal, so the mode says little.
Grouped median and crude mode
When only grouped data are available, the crude mode is the midpoint of the interval with the highest frequency. The median is estimated by interpolation within the interval that contains the (N/2)th score.
Median = L + [(N/2 − F) / f] × i
L
exact lower limit of the median interval
F
cumulative frequency below it
f
frequency of the median interval
i
interval width
≈ 28.67Grouped medianGrouped table of 30 scores, median in 20–29
24.5Crude modeMidpoint of 20–29
Explain
In a grouped table the individual scores are lost, so the median has to be estimated. The formula finds the interval that holds the middle score, assumes the scores are spread evenly within it, and moves into the interval by the right fraction of its width. In the example, 4 scores lie below 19.5 and the middle is the 15th score, so the median lies 11 of the interval's 12 scores into it: 19.5 + (11/12) × 10 ≈ 28.67.
Check yourself
Why is the mean sensitive to extreme scores?
Every score contributes its exact value to the mean.
What percentile is the median?
The 50th percentile.
Which measure of central tendency is appropriate for nominal data?
The mode.
What is the median of 4, 5, 5, 6, 30?
5.
What happens when every score occurs equally often?
There is no mode.
What is the crude mode in grouped data?
The midpoint of the class interval with the highest frequency.
Section 05
Measures of Variability
Very important
11 cards
Variability
Variability describes how widely scores are dispersed around their center. Two distributions can share a mean but differ greatly in spread, and variability also affects the precision of estimates and the power of statistical tests.
The range, highest minus lowest, is the simplest measure. It uses only two scores and is the least stable measure of variability across samples.
The measures run from the range, through the interquartile range and the average deviation, to the variance and standard deviation, which use every score. The coefficient of variation compares spread across different units.
Explain
Two classes can both average 70 marks yet be very different to teach. In one, nearly everyone scores between 67 and 73; in the other, scores run from the 40s to the 90s. The mean alone hides this difference, so a measure of spread is needed alongside it. Spread also matters in research: the more scores vary, the harder it is to detect a real difference between groups.
Quartiles, IQR and the box plot
A percentile is the score at or below which a given percentage of the distribution falls. Q1 is the 25th percentile, Q2 the median, and Q3 the 75th percentile, while deciles divide the distribution into ten equal parts. The IQR (Q3 − Q1) covers the middle 50% of scores, and the quartile deviation, or semi-interquartile range, is half of it.
Tukey's rule flags scores more than 1.5 × IQR below Q1 or above Q3 as potential outliers.
Explain
A box plot shows five things at once: the median, the two quartiles that form the box, and the whiskers that reach the lowest and highest ordinary scores. The box holds the middle half of the scores, so a long box means the middle scores are spread out. A median line off-centre in the box, or one whisker much longer than the other, shows that the distribution is skewed, and points beyond the whiskers mark scores that lie unusually far out.
Quartiles and IQR example
1
Order the scores2, 4, 6, 7, 9, 11, 14, 16, 20
2
Find the median9
3
Split into halvesThe median-of-halves method excludes the overall median
4
Find Q1 and Q3Q1 = 5 from 2, 4, 6, 7; Q3 = 15 from 11, 14, 16, 20
5
Compute spreadIQR = 10; QD = 5
Average deviation
The average deviation, or mean absolute deviation, averages the absolute differences between each score and the mean. It stays in the original units but is rarely used in inferential statistics.
AD = Σ|X − M| / N
M
mean
|X − M|
absolute deviation from the mean
6Mean of 2, 4, 6, 8, 10
12Sum of absolute deviations
2.4Average deviation
Explain
Deviations from the mean always add up to zero, because the scores above and below the mean balance out, so their plain average would always be zero and say nothing. Taking absolute values, ignoring the minus signs, solves this, and the average deviation then reads directly as the typical distance of a score from the mean. In the example, the scores 2, 4, 6, 8 and 10 lie on average 2.4 points from their mean of 6.
Variance and standard deviation
Variance is the average of the squared deviations from the mean. The standard deviation is its square root, which returns the measure to the original units.
Population: σ² = Σ(X − μ)² / N and σ = √σ²
Sample: s² = Σ(X − M)² / (n − 1) and s = √s²
N
population: divide by N
n − 1
sample: Bessel's correction
Deviations from a sample's own mean are, on average, slightly smaller than deviations from μ. Dividing by n − 1 corrects this underestimate and makes sample variance an unbiased estimator of population variance.
Explain
Squaring the deviations also removes the minus signs, and it makes large deviations count much more than small ones. Because squared units, such as marks squared, are hard to picture, the square root brings the result back to the original units, giving the standard deviation. A sample's scores sit a little closer to their own mean than to the true population mean, so dividing by n − 1 instead of n makes the sample variance slightly larger and corrects for this.
Standard deviation: N or n − 1
The sum of squared deviations can be found directly or with the computational form. For the scores 2, 4, 4, 4, 5, 5, 7, 9, the mean is 5.
SS = Σ(X − M)² = ΣX² − (ΣX)² / n
SS
sum of squared deviations
32SS
2σ, treated as a population (N = 8)
≈ 2.14s, treated as a sample (n − 1 = 7)
The choice of denominator follows from whether the scores are the whole group of interest or a sample used to estimate a population.
Common confusion: (ΣX)² and ΣX²
Incorrect(ΣX)² and ΣX² are interchangeable.
Correct(ΣX)² means sum the scores and then square the total, whereas ΣX² means square each score and then sum. For 2, 3, 4 the values are 81 and 29 respectively.
Coefficient of variation
The coefficient of variation expresses the SD as a percentage of the mean, so relative spread can be compared across different units or very different means. It is used for ratio-scale data with a nonzero mean, not for interval scales such as Celsius temperature or IQ scores.
CV = (s / M) × 100
s
sample standard deviation
M
sample mean
Mean
SD
CV
Reaction time
400 ms
60 ms
15%
Height
160 cm
8 cm
5%
Explain
The coefficient of variation answers questions such as whether people differ more in reaction time or in height. The raw standard deviations, 60 ms and 8 cm, cannot be compared because the units differ. Expressed as a percentage of each mean, reaction times vary by 15% and heights by 5%, so reaction times are three times as variable relative to their average.
Comparing variability measures
Uses every score
Units
Typical use
Range
No
Original
Quick, rough description
IQR / QD
No
Original
Skewed data or outliers; reported with the median
Average deviation
Yes
Original
Simple descriptive summary
Variance
Yes
Squared
Statistical theory; ANOVA
Standard deviation
Yes
Original
Standard description of spread; reported with the mean
Coefficient of variation
Yes
None (%)
Relative spread across different units or means
Transforming every score
Mean, median, mode
SD
Variance
Add a constant c
Each increases by c
Unchanged
Unchanged
Multiply by a constant k
Each multiplied by k
Multiplied by |k|
Multiplied by k²
Convert to z-scores
Mean becomes 0
Becomes 1
Becomes 1
None of these changes alters shape or rank order when k is positive. Adding five grace marks to every score raises the mean by five and leaves the SD and skewness unchanged, but the CV changes because the mean rises. Multiplying by a negative constant reverses the order of the scores and the direction of any skew.
Explain
Adding the same amount to every score slides the whole distribution along the scale without changing the gaps between scores. If a teacher adds five grace marks to every student, each score and the mean rise by five, but the spread, the order of the students and the shape stay exactly the same. Multiplying every score by a number, such as doubling marks out of 50 to make them out of 100, stretches the gaps, so the SD is multiplied by the same number and the variance by its square.
Check yourself
Why is the range unstable?
It depends only on the highest and lowest scores, so one extreme score can change it greatly.
What proportion of the ordered distribution does the IQR cover?
The middle 50%.
Why does sample variance divide by n − 1?
Bessel's correction offsets the tendency for deviations from the sample mean to underestimate population variability.
What are the units of variance?
Squared units.
For which data is the coefficient of variation used?
Ratio-scale data with a meaningful zero and a nonzero mean.
What happens to SD if the same constant is added to every score?
It remains unchanged.
What happens to variance if every score is multiplied by k?
Variance is multiplied by k².
Section 06
Distribution Shape and the Normal Curve
Very important
10 cards
Skewness
Distribution shape is described by skewness, which concerns symmetry, and kurtosis, which concerns tail weight. Positive (right) skew has its long tail toward high scores, as with income or reaction times. Negative (left) skew has its long tail toward low scores, as on an easy test.
Positive skew: Mode < Median < Mean. Negative skew: Mean < Median < Mode.
The normal distribution is the reference shape for both: symmetric, with fixed proportions of scores within 1, 2 and 3 SD of the mean. The section ends with ways of checking whether data are normal.
Explain
Skew is named after the tail, not the hump. Income is positively skewed: most people earn moderate amounts and a few earn a great deal, which stretches a long tail to the right. Marks on a very easy test are negatively skewed: most students score high, and a few low scores trail off to the left. In each case the mean is pulled furthest toward the tail, the mode stays at the peak, and the median falls between them.
Pearson's skewness relations
The mean is pulled furthest toward the long tail, the median less so, and the mode stays at the peak. For moderately skewed distributions, Karl Pearson's empirical relation links the three measures.
Mode ≈ 3 Median − 2 Mean
Sk = (Mean − Mode) / SD
Sk = 3(Mean − Median) / SD
Sk
coefficient of skewness
SD
standard deviation
The sign gives the direction of skew: positive for right skew, negative for left skew, and 0 for symmetry.
Explain
Pearson's coefficients turn the gap between the mean and the mode, or between the mean and the median, into a number. If the mean lies above the mode, the tail is on the right and the coefficient is positive; if it lies below, the coefficient is negative. The empirical relation lets the mode be estimated from the mean and median: with a mean of 50 and a median of 48, the mode is about 3 × 48 − 2 × 50 = 44.
The normal distribution
The normal distribution is a theoretical, symmetric, unimodal probability distribution defined by its mean and standard deviation. Its odd-order central moments are zero, its second central moment is the variance, and its kurtosis is the reference point for describing other distributions.
CenterMean = median = mode
Skewness0
TailsAsymptotic: they approach the horizontal axis but do not meet it
AreaThe total area under the curve represents all scores
Explain
Measurements that depend on many small, independent influences, such as height or scores on a well-built ability test, follow the normal curve closely. Because the curve is fully described by its mean and standard deviation, these two numbers tell how scores are spread across the whole range. The tails never touch the axis, so extreme scores are rare but never impossible.
Areas under the normal curve
The interval ±1.96 SD contains almost exactly 95% of scores.
Explain
The 68–95–99.7 pattern turns standard deviations into proportions. On an IQ test with a mean of 100 and an SD of 15, about 68% of people score between 85 and 115, about 95% between 70 and 130, and nearly everyone between 55 and 145. A score of 130 is therefore rare: only about 2% of people score higher.
Normal-curve percentiles
Mean to this point
Within ± this distance
Percentiles at +z and −z
1 SD
34.13%
68.26%
84.13 and 15.87
2 SD
47.72%
95.44%
97.72 and 2.28
3 SD
49.87%
99.74%
99.87 and 0.13
Approximately 68% of scores lie within ±1 SD of the mean, 95% within ±2 SD, and 99.7% within ±3 SD.
Spread measures under normality
For the normal curve, the average deviation is about 0.80σ, roughly 4/5 SD, and the quartile deviation is about 0.6745σ, roughly 2/3 SD.
Quartile deviation0.6745σ
Average deviation≈ 0.80σ
Standard deviation1σ
For the normal curve, QD < AD < SD.
Kurtosis
Kurtosis describes the tails of a distribution compared with the normal curve. A normal distribution is mesokurtic, with a kurtosis of 3. Excess kurtosis subtracts 3, so for the normal curve it is 0.
Greek root
Excess kurtosis
Tails
Mesokurtic
meso-: middle
0
As in the normal distribution
Leptokurtic
lepto-: slender
Positive
Heavier than normal
Platykurtic
platy-: flat
Negative
Lighter than normal
Explain
A leptokurtic distribution looks slender and tall, with more scores piled near the centre and more extreme scores in the tails. A platykurtic distribution looks flat, with scores spread more evenly and fewer extremes. The normal curve sits between the two and is mesokurtic, with a kurtosis of 3.
Common confusion: kurtosis and excess kurtosis
IncorrectThe kurtosis of a normal distribution is zero.
CorrectThe kurtosis of a normal distribution is 3. Excess kurtosis, which subtracts 3, is 0.
Checking normality
Normality can be examined graphically and with formal tests.
Q-Q plotObserved quantiles against those expected under normality; points along a straight reference line indicate normality
HistogramA visual check of the shape
Formal testsShapiro-Wilk, Kolmogorov-Smirnov, and Jarque-Bera, which is based on sample skewness and kurtosis
A scatterplot does not test univariate normality, because it displays the relationship between two variables.
Explain
A Q-Q plot lines up the observed scores against the scores a normal distribution would produce. If the data are normal, the points fall along a straight line; a curved pattern shows skew, and points bending away at the ends show heavy or light tails. The Shapiro-Wilk test asks the same question as a significance test, and a significant result means that the data depart from normality.
Check yourself
What is the ordering of mean, median, and mode in positive skew?
Mode < Median < Mean.
What is the ordering in negative skew?
Mean < Median < Mode.
What percentage of a normal distribution lies within about ±2 SD?
About 95%; the table value is 95.44%.
What proportion lies between the mean and +1 SD?
About 34.13%.
What are normal kurtosis and normal excess kurtosis?
Ordinary kurtosis is 3 and excess kurtosis is 0.
What does a Q-Q plot examine?
Whether observed quantiles lie close to the quantiles expected under a normal distribution.
Is the kurtosis of a normal distribution zero?
No. It is 3; only the excess kurtosis is 0.
Section 07
z-Scores and Standard Scores
Very important
8 cards
z-scores
A z-score expresses a raw score in standard-deviation units. Positive values lie above the mean, negative values below it, and z = 0 lies at the mean.
Population: z = (X − μ) / σ
Sample: z = (X − M) / s
X
raw score
μ or M
mean
σ or s
standard deviation
With M = 70 and SD = 10, a score of 85 has z = (85 − 70) / 10 = +1.5, so it lies 1.5 SD above the mean. A raw score is recovered with X = M + (z × s).
The standard normal distribution is the normal distribution with mean 0 and SD 1. z-scores can be converted to other standard scores, such as T-scores and deviation IQs.
Explain
A z-score answers how unusual a score is within its own group. A score of 85 means little on its own, but z = +1.5 says that it lies one and a half standard deviations above the class average. Because z-scores strip away the original units, scores from tests with different means and spreads can be compared directly.
Comparing scores across tests
z-scores place results from different distributions on a common scale.
Test A
Test B
Raw score
75
68
Mean
60
60
SD
10
4
z
1.5
2.0
The lower raw score on Test B represents the stronger relative performance.
Common confusion: z-scores and the normal distribution
IncorrectConverting scores to z-scores makes their distribution normal.
CorrectConversion gives a mean of 0 and an SD of 1 but leaves the shape and the order of the scores unchanged, so skewed raw scores produce equally skewed z-scores. The standard normal distribution is the normal distribution with mean 0 and SD 1.
Standard-score scales
A z-score converts to another scale with new score = desired mean + (desired SD × z). The number changes with the scale, but the person's relative position is preserved.
z = +2 corresponds to T = 70 and a deviation IQ of 130.
Explain
Raw z-scores are awkward to report: half of them are negative and most have decimals. Other standard scores simply rescale z to a friendlier mean and SD, so a T-score of 70 and a deviation IQ of 130 say the same thing as z = +2: two standard deviations above the mean. To convert, multiply z by the new SD and add the new mean.
Standard scores and their uses
Mean
SD
Typical use
z-score
0
1
Basic standardization
Conventional T-score
50
10
Personality and clinical scales, such as the MMPI
Deviation IQ
100
15
Wechsler scales; the fifth edition of the Stanford-Binet
Stanford-Binet IQ up to the 4th edition
100
16
Older Stanford-Binet editions
Wechsler subtest scaled score
10
3
Individual Wechsler subtests
AGCT standard score
100
20
Army General Classification Test
A conversion always uses the mean and SD of the scale in question.
Stanines
Stanines ("standard nine") divide the normal distribution into nine broad bands. They have a mean of 5 and an SD of about 2, so a stanine is about 5 + 2z, rounded to 1–9. They are used for broad achievement and aptitude reporting.
12%17%20%17%12%
4% Stanine 1
7% Stanine 2
12% Stanine 3
17% Stanine 4
20% Stanine 5
17% Stanine 6
12% Stanine 7
7% Stanine 8
4% Stanine 9
Explain
Stanines group scores into nine broad bands, which keeps reports simple and stops people over-reading tiny differences between scores. The middle band, stanine 5, holds the largest share, 20%, and the bands shrink toward the ends. A student in stanine 7, 8 or 9 is above average, and one in stanine 1, 2 or 3 is below average.
Deviation IQ and percentiles
An IQ score is not a percentile. On a normal distribution with M = 100 and SD = 15, the scores correspond as follows.
z
Approximate percentile
DIQ 85
−1
16th
DIQ 115
+1
84th
DIQ 130
+2
98th
Check yourself
What does z = 0 indicate?
The raw score is at the mean.
What does z = −1 mean?
The score is one standard deviation below the mean.
What happens to mean and SD after conversion to z-scores?
The mean becomes 0 and the SD becomes 1.
Does converting to z-scores make a skewed distribution normal?
No. The shape is unchanged.
What T-score corresponds to z = +2?
70.
What deviation IQ corresponds to z = +2?
130.
Section 08
Probability Foundations
Important
5 cards
Probability
Probability expresses uncertainty on a scale from 0 for an impossible event to 1 for a certain one. When all outcomes are equally likely, it is found by counting outcomes, so the probability of drawing one specified card from a 52-card deck is 1/52.
p(A) = number of favorable outcomes / total number of possible outcomes
p(A)
probability of event A
The total area under a probability distribution is 1. For the normal distribution, the area between two z-scores gives the probability that a randomly selected score falls in that range.
Two rules combine events: the addition rule gives the probability that at least one of them occurs, and the multiplication rule the probability that both occur.
Explain
Probability puts a number on how likely something is. A fair coin has two equally likely outcomes, so the probability of heads is 1/2, or .5. Statistics applies the same idea to areas under a curve: if 34% of a normal distribution lies between the mean and +1 SD, the probability that a randomly chosen score falls there is .34.
Addition rules
The addition rule gives the probability that at least one of two events occurs. Mutually exclusive events cannot occur together.
Condition
Formula
Example
Special addition
Mutually exclusive
p(A or B) = p(A) + p(B)
King or queen = 4/52 + 4/52 = 8/52
General addition
Events can occur together
p(A or B) = p(A) + p(B) − p(A and B)
King or heart = 4/52 + 13/52 − 1/52 = 16/52
Explain
The general rule subtracts the overlap so that it is not counted twice. In a deck of cards the king of hearts is both a king and a heart, so adding the 4 kings and the 13 hearts counts it twice, and 1 card is subtracted, giving 16 cards out of 52. When events cannot happen together, such as drawing a king and a queen in a single draw, there is no overlap and the probabilities simply add.
Multiplication rules
The multiplication rule gives the probability that both events occur. Independent events can occur together, and one does not change the probability of the other.
Condition
Formula
Example
Special multiplication
Independent
p(A and B) = p(A) × p(B)
Two heads = 1/2 × 1/2 = 1/4
General multiplication
Dependent
p(A and B) = p(A) × p(B | A)
Two aces without replacement = 4/52 × 3/51
Explain
Independent events do not affect each other: a coin has no memory, so the second toss is still 1/2 heads whatever happened on the first. Dependent events do: after one ace is drawn and not put back, only 3 aces remain among 51 cards, so the second probability becomes 3/51. The probability that both happen is the first probability multiplied by the second, adjusted for what happened first.
Common confusion: mutually exclusive and independent events
IncorrectMutually exclusive events are independent because they are separate events.
CorrectMutually exclusive events cannot occur together, so p(A and B) = 0. Independent events can occur together, and the occurrence of one does not change the probability of the other. Two mutually exclusive events with nonzero probabilities are therefore dependent.
Check yourself
What is the probability range?
0 to 1.
When can classical probability be found by counting favorable and total outcomes?
When all possible outcomes are equally likely.
Which addition rule applies to mutually exclusive events?
p(A or B) = p(A) + p(B).
Which multiplication rule applies to independent events?
p(A and B) = p(A) × p(B).
What is p(A and B) for mutually exclusive events?
0.
Section 09
Sampling Distributions, Standard Error and the Central Limit Theorem
Important
7 cards
Sampling distributions
A sampling distribution is the distribution of a statistic across all possible samples of a fixed size from the same population. Inferential statistics depend on the fact that sample statistics vary from sample to sample.
Populationmean μ, SD σ
Many samples of size ndrawn repeatedly
One mean from each sampleM₁, M₂, M₃ …
Sampling distribution of meanscenter μ, SD σ/√n
The standard deviation of this distribution is the standard error. It is used to build confidence intervals, and the Central Limit Theorem explains why the sampling distribution of the mean is approximately normal.
Explain
Imagine drawing a sample of 36 students again and again from the same college and recording each sample's mean height. The means would not all be equal; they would form their own distribution, centred on the true population mean. This distribution of sample means is the sampling distribution, and its spread shows how far a single sample mean is likely to be from the truth.
Standard error of the mean
The standard error is the standard deviation of the sampling distribution of means. It describes how much sample means vary across repeated samples and is normally estimated from the sample SD.
σ known: SE = σ / √n
Estimated: SE = s / √n
σ
population standard deviation
s
sample standard deviation
n
sample size
If σ = 15 and n = 36, SE = 15 / 6 = 2.5.
Because n is under a square root, quadrupling the sample size halves the standard error.
Explain
The standard error shrinks with the square root of the sample size, so precision improves quickly at first and then more slowly. Going from 25 to 100 people halves the standard error, but halving it again needs 400 people. Very large samples therefore give only small further gains in precision.
Common confusion: Standard deviation and standard error
IncorrectThe standard deviation and the standard error describe the same variation.
CorrectThe standard deviation describes variation among individual observations and does not shrink systematically as n increases. The standard error describes variation among sample estimates across repeated samples; for the mean it equals SD / √n and decreases as n increases.
Confidence intervals
A confidence interval uses the standard error to express an estimate as a range of plausible values for the parameter. For a normal sampling distribution, a 95% interval for the mean is approximately M ± 1.96 SE.
M = 104
99.1108.9
95% confidence interval
With M = 104 and SE = 2.5, the interval runs from about 99.1 to 108.9. We can be 95% confident that the interval contains μ: across repeated samples, about 95% of intervals built this way would contain it. A narrower interval indicates a more precise estimate, and with a small sample and unknown σ a critical t-value replaces 1.96.
Explain
A confidence interval gives a range instead of a single guess. If a sample of students has a mean score of 104 with a standard error of 2.5, the 95% interval runs from about 99 to 109. A larger sample gives a smaller standard error and therefore a narrower, more precise interval, while a 99% interval, which uses a larger multiplier, is wider than a 95% one.
Central Limit Theorem
The sampling distribution of the sample mean becomes increasingly close to normal as n grows, even when the population is not normal. The theorem concerns the distribution of sample means, not the shape of the raw scores.
CenterThe mean of the sampling distribution equals μ
SpreadIts standard deviation is the standard error, σ/√n
ShapeIt approaches normality as n increases
Near μSample means close to μ are more frequent than sample means far from it
n ≈ 30The usual rule for a sample large enough for the sampling distribution to be approximately normal
Explain
Rolling one die gives a flat distribution: each number from 1 to 6 is equally likely. But the average of several dice rolled together is most often near 3.5 and rarely near 1 or 6, and as more dice are averaged, the distribution of that average looks more and more like a normal curve. The same happens with sample means from any population, which is why normal-curve methods can be used for means even when the raw scores are not normal.
Bootstrap resampling
The bootstrap estimates sampling variability by repeatedly drawing samples with replacement from the observed sample and recalculating the statistic.
1
Start with the observed sample
2
Resample with replacementUsually using the same sample size
3
Recalculate the statistic
4
Repeat many times
5
Use the bootstrap distributionEstimate a standard error or confidence interval
The method applies to many statistics and to metric as well as non-metric data, without assuming a particular population distribution. It is a resampling method rather than simply another rank-based non-parametric test.
Explain
The bootstrap treats the sample as a stand-in for the population. It draws many new samples of the same size from the original one, putting each score back before the next draw, so some scores appear twice and others not at all. The spread of the statistic across these resamples shows how much it would vary from sample to sample, without any assumption about the shape of the population.
Check yourself
What is a sampling distribution?
The distribution of a statistic across all possible samples of a fixed size from the same population.
What does the standard error of the mean describe?
Variation among sample means across repeated samples.
If σ = 15 and n = 36, what is SE?
2.5.
What happens to SE when sample size is quadrupled?
It is halved.
What does a 95% confidence level describe?
Across repeated sampling, about 95% of intervals constructed by the method would contain μ.
What three properties characterize the sampling distribution of the mean?
Its mean equals μ, its SD is σ/√n, and its shape approaches normality as n increases.
What is the defining resampling feature of the bootstrap?
Samples are repeatedly drawn with replacement from the observed sample.
Section 10
Hypothesis Testing, p-Values, Errors and Power
Very important
9 cards
Hypothesis testing
Hypothesis testing compares the observed data with what would be expected under a null hypothesis, following a fixed sequence of decisions.
1
State H₀ and H₁H₀ states that there is no population effect, difference, or association; H₁ states that one is present
2
Set αThe preselected Type I error rate, commonly .05 or .01
3
Compute the test statisticA standardized summary of the evidence, such as z, t, F, or χ²
4
Find p or the critical value
5
Reject H₀ or fail to reject it
The decision can go wrong in two ways, a Type I or a Type II error, and power is the probability of correctly detecting a real effect.
Explain
Hypothesis testing works like a trial. It begins by assuming the null hypothesis, that there is no effect, and asks whether the data would be surprising if that were true. If they would be very surprising, the null hypothesis is rejected; if not, it stands, although it has not been proved.
p-values and the decision rule
A p-value is the probability, assuming H₀ and the statistical model are true, of obtaining the observed result or a result at least as extreme.
Decision
Interpretation
p ≤ α
Reject H₀
The result is statistically significant
p > α
Fail to reject H₀
The result is not statistically significant
Failing to reject H₀ does not prove that there is no effect. Rejecting H₀ does not prove H₁ either, because a significant result still occurs with probability α when H₀ is true. Statistical significance also does not show how large or important an effect is.
Explain
A small p-value means the result would be unlikely if there were really no effect. If a new teaching method seems to raise marks and p = .02, a difference this large would appear by chance only about 2 times in 100 if the method made no difference. Because .02 is below α = .05, the result is called significant.
Common confusion: p-values and the null hypothesis
Incorrectp = .03 means there is a 3% probability that H₀ is true.
Correctp = .03 is the probability of the observed result or a result at least as extreme, assuming H₀ and the statistical model are true.
Critical regions and tails
The critical region contains test-statistic values extreme enough to reject H₀, and its boundary is the critical value. A two-tailed test divides α between both tails, whereas a one-tailed test places all of α in one tail.
At α = .01 the critical values are ±2.58 for a two-tailed test and 2.33 for a one-tailed test. A one-tailed test cannot detect an effect in the opposite direction and is justified only when the direction is predicted before data collection.
Explain
A two-tailed test asks whether a difference exists in either direction, so it splits α between both ends and needs a z beyond ±1.96 at α = .05. A one-tailed test asks about one direction only, so all of α sits in one tail and a z beyond 1.645 is enough. The one-tailed test is more sensitive in the predicted direction but blind to a difference in the other.
Type I and Type II errors
H₀ true
H₀ false
Reject H₀
Type I errorFalse positive; probability α
Correct rejectionProbability 1 − β, the power
Fail to reject H₀
Correct decision
Type II errorFalse negative; probability β
A lower α reduces the risk of a Type I error. A larger n reduces β and increases power without changing α.
Explain
A medical test shows the two errors clearly. Telling a healthy person that they have a disease is a false positive, a Type I error; telling a sick person that they are healthy is a false negative, a Type II error. Lowering α makes false positives rarer, but unless the sample grows, it makes false negatives more common.
Courtroom analogy
A trial begins from the null hypothesis that the defendant is innocent.
Convicting an innocent personType I errorRejecting a true H₀
Acquitting a guilty person for lack of evidenceType II errorFailing to reject a false H₀
A verdict of not guiltyInsufficient evidenceInnocence is not proved, just as failing to reject H₀ does not prove it
Power and the trade-off between α and β
Power = 1 − β, the probability of rejecting H₀ when H₀ is false. Making α more stringent increases β, so moving from α = .05 to α = .01 lowers the risk of a Type I error but raises the risk of a Type II error.
Larger samplesIncrease power
Larger true effectsIncrease power
Lower unexplained variabilityIncreases power
Less stringent αIncreases power
One-tailed testIncreases power when the effect lies in the predicted direction
Explain
Power is the chance that a study detects an effect that really exists. A study with too few participants may miss a real difference simply because the result is not clear enough to pass the significance test. Researchers therefore plan the sample size in advance so that power reaches a target such as .80, an 80% chance of detecting the expected effect.
Cohen's conventional power target
Cohen proposed a power of .80 as a conventional target, which sets β = .20 when α = .05. The ratio β / α = .20 / .05 = 4 treats a Type I error as four times as serious as a Type II error.
β, the Type II error risk.20
α, the Type I error risk.05
Check yourself
What does H₀ state?
That there is no population effect, difference, or association of the form being tested.
When is H₀ rejected using a p-value?
When p ≤ α.
What does p = .03 mean?
Assuming H₀ and the statistical model are true, the probability of the observed result or one at least as extreme is .03.
What is a Type I error?
Rejecting H₀ when H₀ is true; its probability is α.
What is a Type II error?
Failing to reject H₀ when H₀ is false; its probability is β.
How is power related to β?
Power = 1 − β.
What happens to β when α is made more stringent?
β increases.
Section 11
Researchers
1 card
Researchers and contributions
Five researchers gave statistics some of its best-known tools, from de Moivre's normal curve in 1733 to Efron's bootstrap in 1979.
1733Abraham de MoivreDerived the normal curve as an approximation to the binomial distribution
1925R. A. FisherPopularized p = .05 as a convenient threshold for judging significance
1946S. S. StevensProposed the nominal, ordinal, interval, and ratio classification and linked each scale to permissible statistics
1977John TukeyPopularized the box plot in Exploratory Data Analysis and the 1.5 × IQR potential-outlier rule
1979Bradley EfronIntroduced the bootstrap resampling method
Section 12
Previous Year Questions
Mean, median and mode: calculation and choice
Variance, SD and CV calculations
Normal-curve areas and percentiles
Type I and Type II errors, α, β and power
Adding or multiplying every score
Scales of measurement
Skewness and the order of mean, median and mode
z-scores and standard-score conversion
Null hypothesis and significance decisions
Standard error and the Central Limit Theorem
Kurtosis and normality checks
Quartiles, IQR and box plots
Variable types and graphs
Bootstrap resampling
Probability rules
UGC NET1 / 6
The teacher had marks for 50 students in his class. He computed their mean and standard deviation. Considering that the evaluation was too strict. He gave five grace marks to each student. This would
a)Increase mean by five
b)Alter standard deviation in an unpredicted way
c)Change the rank order of the students
d)Change the skewness of the distribution
Section 13
Quick Revision
Quick Revision
Tap any row to reveal the answer
Data and Measurement
Parameter vs statistic
parameter describes a population (μ, σ); statistic describes a sample (M, s)
Scales in order of information
nominal → ordinal → interval → ratio
Ratio vs interval scale
ratio scale has a true zero, so ratios are meaningful
Continuous variable
can take any value within an interval, e.g. height
Histogram vs bar chart
histogram bars touch over continuous intervals; bar-chart bars are separate categories
Central Tendency
Measure for nominal data
mode only
Median with an even n
mean of the two middle scores after ordering
Positive skew (usual unimodal pattern)
Mode < Median < Mean
Negative skew (usual unimodal pattern)
Mean < Median < Mode
Empirical relation
Mode ≈ 3 Median − 2 Mean
Variability
Dispersion measures that use every score
average deviation, variance, SD
Quartile deviation
(Q3 − Q1)/2; half the IQR
Sample variance
uses n−1; population variance uses N
Adding a constant
mean changes by the constant; SD is unchanged
Multiplying by k
mean × k; SD × |k|; variance × k²
CV
100 × SD/mean; most meaningful for ratio-scale data with a meaningful nonzero mean
Normal Curve and Standard Scores
Normal distribution
symmetric, unimodal, asymptotic; mean = median = mode
68-95-99.7
approximately ±1, ±2, ±3 SD around the mean
Area between the mean and +1 SD
about 34%
Percentile at z = +1
about the 84th; z = −1 is about the 16th
Kurtosis of normal
3 ordinary kurtosis; 0 excess kurtosis
z-score
distance from mean in SD units; transformed mean 0, SD 1
Conventional T-score
M = 50, SD = 10; T = 50 + 10z
Wechsler deviation IQ
M = 100, SD = 15; DIQ = 100 + 15z
Sampling and Inference
Standard error
SD of a sampling distribution; SE of mean = σ/√n
CLT
sampling distribution of the mean approaches normal as n grows; n≈30 is a rule of thumb
95% confidence interval for a mean
approximately M ± 1.96 SE
p-value
probability of observed-or-more-extreme data assuming H₀/model
Type I error
reject a true H₀; probability α
Type II error
fail to reject a false H₀; probability β
Power
1 − β; generally increases with larger n and larger true effects