Data Analysis: Descriptive Statistics

Descriptive statistics summarize and organize data to make patterns visible without drawing conclusions beyond the sample. They form the foundation of quantitative analysis in psychological research.

Measures of Central Tendency describe the 'typical' value:

• Mean (x̄) = Σx/n (sum of all values divided by number of values). Most sensitive to extreme scores; used with interval/ratio data.

• Median = middle value when data arranged in order. Robust against outliers; suitable for ordinal, interval, or ratio data.

• Mode = most frequently occurring value. Only measure for nominal data; can be used with all data types.

Measures of Dispersion indicate data spread:

• Range = highest value - lowest value (+ 1 for discrete data). Simple but affected by extreme values.

• Standard Deviation (SD) measures average distance of scores from the mean: σ = √[Σ(x-x̄)²/n]. Shows data clustering; larger SD indicates greater variability.

Key Distinctions: Population parameters use σ (sigma), while sample statistics use s with (n-1) denominator for unbiased estimation.

Normal Distribution: Bell-shaped curve where mean = median = mode. Approximately 68% of data falls within ±1 SD, 95% within ±2 SD, and 99.7% within ±3 SD.

Cambridge Definition: Descriptive statistics are mathematical techniques for organizing, summarizing, and presenting quantitative data in meaningful ways.

Data Types Matter: Nominal (categories), ordinal (ranked), interval (equal intervals, no true zero), ratio (equal intervals with true zero) determine appropriate statistical measures.

Think of descriptive statistics as a psychological snapshot that captures essential features of data without the full detail.

Real-World Application: Sleep Study

Imagine researching whether blue light affects sleep duration. You collect hours slept from 30 participants:

Without descriptive statistics, you'd have 30 individual numbers—overwhelming and meaningless. With them, you might report: "Mean sleep = 6.8 hours (SD = 1.2), indicating most participants slept between 5.6-8.0 hours."

The Restaurant Analogy:

• Mean is like average bill per table—useful but distorted if one group orders champagne (outlier) • Median is the middle bill—better represents typical spending • Mode is the most common order—shows what's genuinely popular • Standard deviation shows whether all tables spend similarly or wildly differently

Clinical Psychology Example

When evaluating depression treatment, researchers measure symptom scores before/after therapy:

• Mean change shows overall effectiveness
• SD reveals consistency—low SD means treatment works similarly for most; high SD suggests it helps some greatly, others minimally
• Median helps when few participants show dramatic improvement (positive skew)

Memory Research Application

Studying recall accuracy: Mode identifies most common error type, mean shows average performance, SD indicates individual differences. A bimodal distribution (two modes) might reveal two distinct participant groups—perhaps different learning strategies.

Professional Context: NHS psychologists use descriptive statistics in clinical audits to track patient outcomes, comparing their service against national benchmarks.

Descriptive statistics transform raw data into actionable insights, enabling psychologists to identify patterns, compare groups, and communicate findings effectively to stakeholders who lack statistical training.

Example 1: Complete Analysis (6 marks)

Question: Ten participants completed a memory test. Scores: 12, 15, 18, 18, 20, 22, 24, 24, 24, 28. Calculate mean, median, mode, range, and standard deviation.

SOLUTION:

Mean: Σx/n = (12+15+18+18+20+22+24+24+24+28)/10 = 205/10 = 20.5

Median: Already ordered. Two middle values (5th & 6th) = (20+22)/2 = 21

Mode: 24 appears three times (most frequent) = 24

Range: 28 - 12 = 16 (or 17 for discrete data)

Standard Deviation:

1. Calculate deviations: (12-20.5)², (15-20.5)², etc.
2. Squared deviations: 72.25, 30.25, 6.25, 6.25, 0.25, 2.25, 12.25, 12.25, 12.25, 56.25
3. Σ(x-x̄)² = 210.5
4. σ = √(210.5/10) = √21.05 = 4.59

Examiner Note: Show all working; marks awarded for process, not just final answer.

Example 2: Comparison Question (4 marks)

Question: Group A (control): Mean = 15, SD = 2.1. Group B (experimental): Mean = 22, SD = 8.3. What do these statistics suggest about the intervention's effect?

SOLUTION:

The experimental group shows higher mean performance (22 vs 15), suggesting the intervention improved scores by 7 points on average. However, the much larger SD (8.3 vs 2.1) indicates greater variability—the intervention helped some participants considerably but may have been less effective for others. The control group's low SD shows consistent performance, while the experimental group's high SD suggests individual differences in response to treatment.

Examiner Note: Always interpret both central tendency and dispersion; discuss implications for the research question.