descriptive statistics

Purpose of Descriptive Statistics

• Descriptive statistics summarise and describe data so that results are easier to interpret.

• They do not test hypotheses — they simply describe what the data show.

• Used to:

  • Identify central tendency (typical score)

  • Measure dispersion (how spread out the data is)

  • Express data as percentages

  • Describe relationships (correlations)

measures of central tendancy

🟢 Mean

• Add up all scores and divide by the number of scores.
  → e.g., (5 + 7 + 8) ÷ 3 = 6.7

• Uses all data values → sensitive measure.

✅ Advantages:

Most accurate measure (uses all data).

• Sensitive to small changes in scores.

Disadvantages:

Affected by extreme scores (outliers).

• Can give a misleading impression if data is skewed.

Use when: Data are interval or ratio and normally distributed.

🟡 Median

• The middle value when scores are ranked in order.
  → e.g., 3, 5, 7 → median = 5

• If even number of scores, take the average of the two middle values.

• ✅ Advantages:

Unaffected by extreme scores (good for skewed data).

• Easy to calculate.

⚠ Disadvantages:

Doesn’t consider all values (less representative).

• Less precise for small samples.

Use when: Data is ordinal or skewed.

🔵 Mode

• The most frequent value in the data set.
  → e.g., 2, 4, 4, 6 → mode = 4

• Can be used with any type of data (nominal, ordinal, interval).

✅ Advantages:

Only measure that can be used with nominal data.

• Not affected by extreme scores.

⚠ Disadvantages:

Can be unrepresentative if multiple modes or no clear mode.

• Doesn’t use all data values.

Use when: Data are categorical or when you need the most common value.

measures of dispersion

🔹 Range

• Simplest measure of spread.

• Calculated as:
  → Highest score − Lowest score (often +1 to include both ends of the range).

✅ Advantages:

Quick and easy to calculate.

⚠ Disadvantages:

Affected by extreme scores.

• Doesn’t show distribution of all scores (only top and bottom).

Use when: Data are ordinal, interval, or ratio, and a simple estimate of spread is enough.

Standard Deviation (SD)

• Shows the average distance of each score from the mean.

• The larger the SD, the more spread out the scores are.

• The smaller the SD, the more consistent the scores are (clustered around the mean).

✅ Advantages:

• Most accurate and informative measure of spread.

• Considers all data values.

⚠ Disadvantages:

• Complicated to calculate.

• Can be misleading if data not normally distributed.

Use when: Data are interval or ratio and distribution is roughly normal.

Interpreting Standard Deviation

• Small SD: scores are close to mean → less variation → consistent results.

• Large SD: scores vary widely → more variation → less consistency.

Percentages

• Express proportions or comparisons of data as percentages.

• Calculated as:
  → (score or frequency ÷ total) × 100

✅ Advantages:

• Easy to understand and compare.

• Standardises results (out of 100).

⚠ Disadvantages:

• Can oversimplify data.

• Doesn’t show variability or spread.

Use when: Presenting data in tables, bar charts, or written summaries.

Correlations

• A measure of the relationship between two variables.

• Described using a correlation coefficient (r) between −1.0 and +1.0.

Interpretation:

• +1.0 → Perfect positive correlation (as one increases, so does the other).

• −1.0 → Perfect negative correlation (as one increases, the other decreases).

• 0 → No correlation (no relationship).

✅ Advantages:

• Shows strength and direction of a relationship.

• Useful for predicting trends.

⚠ Disadvantages:

• Correlation ≠ causation — can’t show cause and effect.

• May be affected by third variables (confounding factors).the