Descriptive Statistics and Distributions

Descriptive statistics summarise a set of data so that a pattern can be seen at a glance instead of reading through raw scores. In AQA Psychology they fall into two jobs: describing the typical value with a measure of central tendency, and describing the spread with a measure of dispersion.

• Central tendency answers "what is a typical score?" — the mean, median and mode.
• Dispersion answers "how spread out are the scores?" — the range and standard deviation.

At least 10% of the marks on the A-level assess mathematical skills, and questions here often give you a small data set and ask you to calculate a value, then justify why that particular statistic was appropriate. The choice of statistic depends on the level of measurement of the data, so the two ideas are linked.

A full answer usually needs both: a measure of central tendency to give the typical value and a measure of dispersion to show how consistent the scores are around it.

Each measure of central tendency finds a "typical" score in a different way, and each has a situation where it is the right choice. The table below is the core knowledge for this topic.

Measure	How to find it	Best for	Strength	Weakness
Mean	Add all scores, divide by the number of scores ($n$)	Interval data	Uses every value, so it is the most sensitive	Distorted by extreme scores (anomalies)
Median	The middle value when scores are in order (average the two middle values if $n$ is even)	Ordinal data	Not affected by extreme scores	Ignores the exact value of most scores
Mode	The most frequently occurring value	Nominal (categorical) data	The only measure usable with categories	A set can have no mode or several modes

The mean is calculated as:

$\bar{x}=\frac{\sum x}{n}$

where $\sum x$ is the sum of all the scores and $n$ is how many scores there are.

Match the statistic to the data. Use the mean for interval data with no anomalies, the median for ordinal data or when there are extreme scores, and the mode for nominal (categorical) data.

Take this data set of scores: 4, 7, 7, 9, 13. Work out all three measures step by step.

Mean — add every score, then divide by $n$:

$\bar{x}=\frac{4+7+7+9+13}{5}=\frac{40}{5}=8$

Median — the scores are already in order (4, 7, 7, 9, 13). With $n=5$ (odd), the middle value is the 3rd score:

$4,7,7,9,13\Rightarrow \text{median}=7$

Mode — count how often each value appears. The value 7 occurs twice; every other value occurs once, so the mode is 7.

If the count were even, you would average the two middle values. For 3, 5, 8, 10 the two middle scores are 5 and 8:

$\text{median}=\frac{5+8}{2}=6.5$

Always put the scores in order before taking the median. The middle of an unordered list is meaningless.

The mean's biggest weakness is that a single anomaly drags it away from the typical score. This is exactly why the level of measurement and the shape of the data decide which statistic is fair.

Start again with 4, 7, 7, 9, 13 (mean = 8, median = 7). Now add one extreme score, 60, to give 4, 7, 7, 9, 13, 60:

$\bar{x}=\frac{4+7+7+9+13+60}{6}=\frac{100}{6}\approx 16.7$

The mean has jumped from 8 to about 16.7, yet only one value changed. The median barely moves. With $n=6$ the two middle values are the 3rd and 4th scores (7 and 9):

$\text{median}=\frac{7+9}{2}=8$

The median stays at 8 — far more representative of the bulk of the scores. This is why, when a data set contains extreme scores, the median is the more honest measure of central tendency.

If a data set has anomalies, prefer the median. The mean is pulled toward the extreme value and stops describing the typical score.

Two data sets can share the same mean but look completely different because their scores are more or less spread out. Measures of dispersion capture that spread.

The range is the simplest: subtract the lowest score from the highest.

$\text{range}=\text{highest}-\text{lowest}$

Worked example — for 4, 7, 7, 9, 13 the highest score is 13 and the lowest is 4:

$\text{range}=13-4=9$

(Some textbooks add 1 to allow for measurement rounding, giving $13-4+1=10$; AQA accepts the simple difference.) The range is quick to calculate but uses only two values, so a single extreme score distorts it and it ignores everything in between.

The standard deviation (SD) measures the average distance of the scores from the mean. It is more precise than the range because it uses every score, not just the two extremes.

• A larger SD means the scores are more spread out — greater variability.
• A smaller SD means the scores cluster tightly around the mean — the group is more consistent.

Standard deviation is more precise than the range because it takes every score into account. Interpreting the size of an SD is the skill AQA tests — a bigger SD simply means more spread, never that the mean is "wrong". The spec requires you to calculate the range, but only to interpret standard deviation, not derive its formula.

Percentages let you compare figures fairly even when the totals differ. To turn a proportion into a percentage, divide the part by the whole and multiply by 100.

$\text{percentage}=\frac{\text{part}}{\text{whole}}\times 100$

Worked example — 18 out of 24 participants recalled a word list correctly:

$\frac{18}{24}\times 100=0.75\times 100=75\%$

Second example — in a different group, 15 out of 20 recalled the list:

$\frac{15}{20}\times 100=0.75\times 100=75\%$

Both groups scored 75%, even though the raw counts (18 and 15) and the totals (24 and 20) were different. That is the value of a percentage: it standardises to "out of 100" so results from samples of different sizes can be compared directly.

To find a percentage: part ÷ whole × 100. Converting raw scores to percentages is the standard way to compare two groups of unequal size on a level footing.

If you plot a large data set as a frequency graph, its shape tells you how the scores are distributed. AQA asks you to recognise three shapes and know where the mean, median and mode sit in each.

A normal distribution is a symmetrical, bell-shaped curve. Most scores cluster around the middle, with fewer scores toward each extreme. Because it is symmetrical, the mean, median and mode all fall at the same central point. Many human characteristics approximate this, such as IQ.

A skewed distribution is asymmetrical — it has one long tail:

• Positive skew — a long tail to the right (toward high scores). Most scores are low, and the few high outliers pull the mean up. A very difficult test produces this: most students score low, so the peak (mode) sits on the left. The order from low to high is mode < median < mean.
• Negative skew — a long tail to the left (toward low scores). Most scores are high, and the few low outliers pull the mean down. A very easy test produces this: most students score high, so the peak sits on the right. The order from low to high is mean < median < mode.

Distribution	Shape	Where the scores cluster	Order of measures (low → high)
Normal	Symmetrical bell	Around the centre	mean = median = mode
Positive skew	Long tail to the right	Most scores low	mode < median < mean
Negative skew	Long tail to the left	Most scores high	mean < median < mode

The mean is always dragged toward the tail. In a positive skew the tail is on the right, so the mean is the highest of the three; in a negative skew the tail is on the left, so the mean is the lowest.

1. Using the mean when the data is heavily skewed

If a data set has extreme scores or a strong skew, the mean is dragged toward the outliers and no longer represents a typical score. In that case the median is the appropriate measure of central tendency. Choosing the mean here and defending it costs the justification marks.

2. Forgetting to order the scores before taking the median

The median is the middle of the scores in size order. Taking the middle item of an unsorted list gives the wrong value. Always sort first, then find the middle (or average the two middle values if the count is even).

3. Mixing up positive and negative skew

Name the skew by the direction of the tail, not where the hump is. A tail pointing right (toward high scores) is a positive skew from a difficult test; a tail pointing left is a negative skew from an easy test.

4. Saying a large standard deviation means the mean is wrong

A large standard deviation means the scores are widely spread around the mean — more variability in the group. It says nothing about the accuracy of the mean itself. Interpret a large SD as "inconsistent scores", not "incorrect average".

5. Confusing the range with the standard deviation

Feature	Range	Standard deviation
Scores used	Only the highest and lowest	Every score
Precision	Low — distorted by one extreme	High — reflects the whole set
What it shows	Total width of the data	Average distance from the mean

The range is a rough, two-value measure; the standard deviation is a precise measure using every score. Do not treat them as interchangeable.