Descriptive Statistics for Psychology
Learning Objectives
- Calculate mean, median, and mode from a raw data set and choose the most appropriate measure for a given distribution.
- Calculate range, variance, and standard deviation, and explain what each tells you about spread.
- Identify normal, skewed, and bimodal distributions and explain what shape reveals about the data.
- Explain how outliers and skew affect the mean but not the median.
- Apply descriptive statistics to real psychological data (test scores, reaction times, survey responses).
Quick Answer
Descriptive statistics are the tools psychologists use to summarize a data set into a few meaningful numbers instead of a long list of raw scores. They fall into two families: measures of central tendency (mean, median, mode), which describe the "typical" score, and measures of variability (range, variance, standard deviation), which describe how spread out the scores are. Together with a picture of the distribution's shape (normal, skewed, or bimodal), descriptive statistics let a researcher say, in one sentence, what an entire data set looks like — the first and essential step before any inferential test can be run.
Central Tendency: Three Ways to Find "Typical"
Mean (μ or x̄) — the arithmetic average. Formula: μ = Σx / n
Worked example — five students' quiz scores: 85, 90, 78, 92, 88 μ = (85 + 90 + 78 + 92 + 88) / 5 = 433 / 5 = 86.6
Median (Mdn) — the middle score when data is ordered. Sort: 78, 85, 88, 90, 92 → the middle value (3rd of 5) is 88. When n is even, average the two middle values.
Mode — the most frequent score. In the example above, every score is unique, so there is no mode. Mode becomes useful with larger, repeating data sets, and it's the only central tendency measure valid for nominal data (e.g., most common diagnosis in a clinic).
Which one should you use? It depends on the shape of the distribution:
- Symmetric, no outliers → mean is most informative (uses all the data).
- Skewed or has outliers → median is more representative (resistant to extreme values).
- Categorical data → mode is the only option.
Variability: How Spread Out Is the Data?
Two scores can share the same mean but tell very different stories depending on how spread out they are. Compare Class A: 84, 85, 86, 85, 85 (mean 85) with Class B: 60, 100, 70, 95, 100 (mean 85). Same average, wildly different consistency — variability measures capture that difference.
Range = highest − lowest. Class A range = 86 − 84 = 2. Class B range = 100 − 60 = 40.
Standard deviation (σ for population, s for sample) — the typical distance of each score from the mean. Formula for a sample: s = √[Σ(x − x̄)² / (n − 1)]
Worked example using the quiz scores (85, 90, 78, 92, 88; mean = 86.6):
| Score | Deviation (x − x̄) | Squared deviation |
|---|---|---|
| 85 | −1.6 | 2.56 |
| 90 | 3.4 | 11.56 |
| 78 | −8.6 | 73.96 |
| 92 | 5.4 | 29.16 |
| 88 | 1.4 | 1.96 |
Sum of squared deviations = 2.56+11.56+73.96+29.16+1.96 = 119.2 Variance = 119.2 / (5−1) = 29.8 Standard deviation = √29.8 ≈ 5.46
So a typical score in this class sits about 5.5 points from the mean of 86.6 — a moderate, unremarkable spread for a 100-point test.
Why divide by n−1 and not n? Using n−1 (called Bessel's correction) corrects for the fact that a sample tends to underestimate the true population variability; it gives a slightly larger, more honest estimate when generalizing beyond the sample.
Shape of the Distribution
The shape of a distribution tells you whether mean and median will agree, and which measures are trustworthy.
- Normal distribution: symmetric, bell-shaped; many psychological traits (IQ, height) approximate this shape. Mean, median, and mode roughly coincide.
- Positively (right) skewed: a few unusually high scores drag the mean above the median. Example: reaction times often skew right because most responses cluster fast, but a few very slow outlier trials pull the tail out.
- Negatively (left) skewed: a few very low scores drag the mean below the median. Example: scores on an easy exam where most students do well but a few fail badly.
- Bimodal: two peaks, often signaling two distinct subgroups mixed in one data set — e.g., reaction times combining both a "correct answer" group and a "guessing" group.
Real-World Application
When a clinical psychologist reports "the average score on the depression inventory was 18 (SD = 6)," a colleague immediately knows two things: the typical severity level, and how much clients varied around it. A small SD signals a homogeneous group (useful for reducing variability in a therapy trial); a large SD signals a very mixed group, which might mean the sample includes both mild and severe cases that should be analyzed separately.
Key Terms
| Term | Definition |
|---|---|
| Central tendency | A single value representing the "typical" or middle score in a data set |
| Mean | The arithmetic average; sum of scores divided by number of scores |
| Median | The middle score in an ordered data set |
| Mode | The most frequently occurring score |
| Range | Highest score minus lowest score |
| Variance | The average of squared deviations from the mean |
| Standard deviation | The square root of variance; typical distance of scores from the mean |
| Skew | Asymmetry in a distribution, with a longer tail on one side |
| Bimodal distribution | A distribution with two distinct peaks |
Common Mistakes
-
Misconception: "The mean is always the best measure of central tendency." Why it's wrong: The mean is pulled toward outliers and skew, which can make it a poor representation of a "typical" case. Correct: For skewed data (income, reaction time, hospital stay length), the median usually gives a more accurate sense of the typical value.
-
Misconception: A standard deviation of zero or a very small SD means "something is wrong with the data." Why it's wrong: A small SD just means scores are tightly clustered — which can happen legitimately, e.g., everyone in a highly trained group performing similarly. Correct: Interpret SD relative to context; small SD indicates consistency, not necessarily an error, though it's worth double-checking data entry when SD is unexpectedly zero.
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Misconception: Range and standard deviation measure the same thing, so you only need to report one. Why it's wrong: Range uses only the two extreme scores and ignores everything in between, making it highly sensitive to a single outlier. Correct: Standard deviation uses every score in the data set, giving a much more stable and informative measure of overall spread.
Comparison and Connections
| Measure | Uses all data? | Sensitive to outliers? | Best for |
|---|---|---|---|
| Mean | Yes | Very sensitive | Symmetric, interval/ratio data |
| Median | No (only middle value) | Resistant | Skewed data, ordinal data |
| Mode | No (only most frequent) | Resistant | Nominal/categorical data |
| Range | No (only extremes) | Very sensitive | Quick, rough sense of spread |
| Standard deviation | Yes | Sensitive | Precise measure of spread for interval/ratio data |
Practice Questions
Recall
- Write the formula for the mean and explain each symbol. Answer guidance: μ = Σx / n; Σx = sum of all scores, n = number of scores.
- What does a large standard deviation indicate about a data set? Answer guidance: Scores are widely spread out from the mean; the group is heterogeneous.
Understanding 3. Explain why the mean and median differ in a skewed distribution. Answer guidance: The mean is pulled toward the tail (extreme scores) because it uses every value in its calculation; the median only reflects the middle position and ignores how extreme the tail values are. 4. Why is standard deviation generally preferred over range as a measure of variability? Answer guidance: SD uses all data points and reflects overall spread, while range depends only on the two most extreme scores and can be distorted by a single outlier.
Application 5. Test scores for 6 students: 55, 60, 58, 95, 62, 59. Calculate the mean and median. Which better represents the "typical" student, and why? Answer guidance: Mean = (55+60+58+95+62+59)/6 = 389/6 ≈ 64.8; ordered data (55,58,59,60,62,95), median = (59+60)/2 = 59.5. The median better represents typical performance since 95 is an outlier inflating the mean. 6. A researcher reports Group A: mean = 50, SD = 2; Group B: mean = 50, SD = 15. What does this tell you about the two groups despite having identical means? Answer guidance: Group A is much more homogeneous/consistent; Group B has far more individual variation even though the average performance is the same.
Analysis 7. A distribution of hospital stay lengths (in days) is strongly right-skewed. Explain which measure of central tendency a hospital administrator should report to the public, and why the alternative could mislead. Answer guidance: Report the median, since a few very long stays would inflate the mean and misrepresent the typical patient's stay length; using the mean could make average care duration look longer than it typically is. 8. Compare a normal distribution to a bimodal distribution in terms of what each shape suggests about the underlying sample. Answer guidance: A normal distribution suggests one homogeneous population varying by chance around a central value; a bimodal distribution suggests two distinct subgroups may be mixed together (e.g., two different strategies or populations), which analysis should probably separate.
FAQ
Why do we need both central tendency and variability measures? Central tendency tells you where the "middle" of the data is, but says nothing about how consistent or spread out the scores are. Two data sets can share a mean yet look completely different once you examine their spread.
When should I report the mean versus the median? Report the mean for roughly symmetric, outlier-free interval/ratio data. Report the median when data is skewed or contains outliers, since it better reflects the "typical" score.
Is a normal distribution the same as a "good" distribution? Not necessarily "good" — it just means many psychological and biological measures naturally distribute this way due to the combined influence of many small random factors. It's common, not inherently superior.
What causes a bimodal distribution in psychological data? Usually a mixture of two distinct subgroups within the sample — for example, combining scores from two different experimental conditions, or two age cohorts, into a single data set.
Do I need to calculate standard deviation by hand on an exam? Many exams expect you to know the formula and steps (deviations, squaring, averaging, square-rooting) and to interpret the result, even if statistical software does the heavy computation in practice.
Quick Revision
- Central tendency measures: mean (average), median (middle), mode (most frequent).
- Mean is sensitive to outliers and skew; median is resistant; mode fits categorical data.
- Variability measures: range (max − min), variance (average squared deviation), standard deviation (√variance).
- Standard deviation uses every score and gives the most reliable measure of spread.
- Divide by n − 1 (not n) when calculating sample variance/SD — this is Bessel's correction.
- Normal distribution: symmetric, bell-shaped, mean ≈ median ≈ mode.
- Positive (right) skew: mean > median, tail points right (e.g., reaction times).
- Negative (left) skew: mean < median, tail points left (e.g., an easy exam).
- Bimodal distribution: two peaks, often signaling two subgroups in the data.
- Descriptive statistics summarize a sample; they do not allow generalizing to a population — that requires inferential statistics.
Related Topics
Prerequisites: Introduction to Statistics (populations, samples, levels of measurement).
Related: Data Interpretation, Correlation and Regression.
Next: Inferential Statistics, where sample descriptive statistics become the basis for generalizing to a population.