6 Educational Statistics · 6.1 Statistics for teachers
6.1.1 Statistics in the classroom: averages, spread and correlation of scores
A teacher first arranges a class's scores (array, frequency table, graph), sums them up with an average (mean, median or mode), shows how spread out they are (range, quartile deviation, mean deviation, variance, standard deviation), places a pupil in the group with percentiles and standard scores, and uses correlation (−1 to +1) to see whether two sets of scores move together.
- APPSC Paper I
- Standard level
- 16 min read
- Free
Core facts
- Measures of central tendency (averages): mean, median, mode. C.V. Good: the tendency of scores to cluster about a point on the scale.
- Mean = sum of scores ÷ number of scores (M = ΣX ÷ N); the mathematical (arithmetic) average; uses every score; pulled by extreme scores.
- Median = the middle score of scores arranged in order: half the scores lie above it and half below; the positional average; not affected by extreme scores; = 50th percentile = Q₂.
- Mode = the score that occurs most often; the only average for nominal data (categories such as blood group, religion, favourite subject).
- Empirical relation (Karl Pearson; moderately skewed data): Mode = 3 Median − 2 Mean.
- Combined mean of two groups = (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂).
- Measures of variability (dispersion, spread, scatter): range, quartile deviation, mean deviation, standard deviation.
- Range = highest score − lowest score (quickest, roughest; depends on two scores only).
- Quartile deviation (Q) = (Q₃ − Q₁) ÷ 2, the semi-interquartile range; goes with the median.
- Mean deviation = average of the deviations from the mean, signs ignored.
- Variance = the average of the squared differences of the scores from the mean (σ²).
- Standard deviation (SD, σ) = square root of the variance; the typical (standard) distance of scores from the mean; the most stable measure of spread; goes with the mean. Term coined by Karl Pearson (1893 lecture, in print 1894).
- Percentile (Pₖ): the score below which k% of the group falls; percentile rank (PR): the percentage of the group scoring below a given score.
- z-score = (X − Mean) ÷ SD (mean 0, SD 1); T-score = 50 + 10z (mean 50, SD 10).
- Normal curve: mean = median = mode; 68.26% of cases within ±1 SD, 95.44% within ±2 SD, 99.73% within ±3 SD.
- Ogive = the cumulative frequency (cumulative percentage) graph; median and percentiles are read from it.
- Correlation = the statistical tool for the relationship between two variables; coefficient from −1 to +1.
- Positive correlation: both rise together and fall together; negative: as one increases the other decreases; zero: no relationship.
- Karl Pearson: product-moment correlation r; Charles Spearman: rank correlation ρ (rho) = 1 − 6ΣD² ÷ [N(N² − 1)]. Francis Galton first gave the idea of correlation ("co-relation") and regression.
Explained
1. Organising scores before averaging
- Array: scores arranged in ascending or descending order.
- Grouped frequency distribution: scores put into class intervals (usually 3, 5 or 10 units wide; about 5 to 15 classes) with tallies and frequencies; total of frequencies = N.
- Exact limits of the class 36–39 are 35.5–39.5; its midpoint is 37.5.
| Graph | How it is drawn | Used for |
|---|---|---|
| Histogram | adjoining bars: base = class interval, height = frequency | shape of the distribution, mode |
| Frequency polygon | frequencies plotted at class midpoints, joined by lines | comparing two distributions |
| Ogive | cumulative % plotted at upper exact limits | median, quartiles, percentiles |
| Pie chart | sector angle = (part ÷ total) × 360° | shares of a whole |
Example: 45 of 180 hostel pupils come from tribal areas, so their sector is 45 ÷ 180 × 360° = 90°.
2. Which average when
| Average | Type | Best for | Weakness |
|---|---|---|---|
| Mean | mathematical | interval data, further work (SD, r); most stable from sample to sample | pulled by extreme scores |
| Median | positional | skewed scores, ranks and letter grades, open-ended classes ("60 and above") | ignores the size of other scores |
| Mode | most frequent | nominal data, "most popular" (one uniform size for most pupils) | may be absent, or two (bimodal) or more |
- Two uses of an average: to describe a group as a whole and to compare groups on the same test.
- Properties of the mean: the sum of deviations from the mean is always zero; the sum of squared deviations is least when taken from the mean.
- The mode is the quickest and roughest average; it can be read from the peak of a histogram.
- The mean is the centre of gravity (balance point) of the scores: the deviations on either side balance out.
- Correcting a wrong entry: 30 pupils, mean 45, but one score of 72 was entered as 42. Correct total = 1350 − 42 + 72 = 1380, so the correct mean = 1380 ÷ 30 = 46.
3. Worked example: one class test (10 pupils, out of 25)
Scores: 12, 15, 9, 18, 15, 11, 20, 15, 13, 12
- Mean = 140 ÷ 10 = 14.
- Median: in order 9, 11, 12, 12, 13, 15, 15, 15, 18, 20. With 10 (even) scores the median is the average of the 5th and 6th: (13 + 15) ÷ 2 = 14.
- Mode = 15 (three times).
- Range = 20 − 9 = 11.
- Percentile rank of 18: 8 of the 10 pupils scored below 18, so PR = 8 ÷ 10 × 100 = 80 (some books also add half of the pupils with the same score).
4. Median of raw scores (odd number). 17, 19, 22, 24, 26, 30, 32, 38, 40: nine scores, already in order, so the median is the 5th score = 26. Rule: median = the ((N + 1) ÷ 2)th score after arranging.
5. Formulas for grouped data
- Mean, long method: M = Σfx ÷ N (x = class midpoint). Short (assumed-mean) method: M = A + (Σfx′ ÷ N) × i.
- Median: Md = l + [(N/2 − F) ÷ f] × i, where l = exact lower limit of the median class, F = cumulative frequency below that class, f = its frequency, i = class width. Example: l = 39.5, N = 50, F = 18, f = 12, i = 5 gives 39.5 + (7 ÷ 12) × 5 ≈ 42.4.
- Q₁ is found the same way at N/4, Q₃ at 3N/4. Q₁ = 25th percentile, Q₃ = 75th percentile; Q₃ − Q₁ (interquartile range) covers the middle 50% of scores.
- Empirical mode: mean 40, median 38 → mode = 3 × 38 − 2 × 40 = 34.
6. Combined mean. Section A: 15 pupils, mean 8; section B: 10 pupils, mean 6. Combined mean = (15 × 8 + 10 × 6) ÷ (15 + 10) = 180 ÷ 25 = 7.2. (Not the simple average 7, because the groups are of different sizes.)
7. Spread in a small set. Scores 4, 6, 8, 10, 12: mean = 8; deviations −4, −2, 0, 2, 4.
- Mean deviation = (4 + 2 + 0 + 2 + 4) ÷ 5 = 2.4 (signs are ignored because the signed deviations always add to zero).
- Squares 16, 4, 0, 4, 16; sum of squares = 40. Variance = 40 ÷ 5 = 8; SD = √8 ≈ 2.83. If SD = 6, variance = 36.
- Σd², the total of the squared deviations from the mean, is called the sum of squares; mean deviation is also called average deviation (AD).
- Variance is in squared units (marks²); SD is in the original units, which is why SD is reported. In testing, total variance = true variance + error variance.
| Measure | Formula | Goes with |
|---|---|---|
| Range | H − L | quick look |
| Quartile deviation | (Q₃ − Q₁) ÷ 2 | median, open-ended classes |
| Mean deviation | Σd (signs ignored) ÷ N | rarely used now |
| Standard deviation | √(Σd² ÷ N), deviations from the mean | mean, normal data |
- In a normal distribution: QD < MD < SD (about 0.67σ, 0.80σ and σ).
- SD = 0 only when all scores are equal.
8. Reading the SD. Two sections have the same mean of 60. Section A has SD 5 (pupils are alike: homogeneous, teach as one group); section B has SD 15 (a wide spread: heterogeneous, plan remedial and enrichment groups). This is why an average alone cannot compare two classes.
9. Effect of changing every score. Adding the same number (5 grace marks) to every score raises the mean, median and mode by 5 but leaves the range and SD unchanged. Multiplying every score by k multiplies the mean and SD by k and the variance by k² (mean 12, every score × 2 then + 3 → new mean 27).
10. Skewed scores. A very easy test piles scores at the top (negatively skewed: mean < median < mode); a very hard test piles them at the bottom (positively skewed: mean > median > mode). In skewed data the median is the safer average.
11. Comparing consistency: coefficient of variation (Karl Pearson) = (SD ÷ mean) × 100, a relative (unit-free) measure. Class A: mean 60, SD 12 → CV 20%; class B: mean 40, SD 10 → CV 25%. Class A is more consistent (lower CV). Range, Q, MD and SD are absolute measures, in score units.
12. Standard scores
| Scale | Mean | SD |
|---|---|---|
| z-score | 0 | 1 |
| T-score | 50 | 10 |
| Stanine (nine bands) | 5 | about 2 |
| Deviation IQ (Wechsler) | 100 | 15 |
Mean 50, SD 10, pupil scores 65 → z = (65 − 50) ÷ 10 = 1.5, T = 65. Standard scores have equal units; percentile units are unequal along the scale.
13. Correlation in school
| Pair of variables | Likely correlation |
|---|---|
| intelligence and school achievement; height and weight | positive |
| hours of TV and marks; typing practice and mistakes | negative |
| roll number and marks; height and marks in English | about zero |
- Kinds of correlation: positive, negative, zero; linear and curvilinear; simple, partial and multiple. ("Cumulative correlation" is not a kind.)
- Correlation needs two sets of scores for the same pupils; r gives the degree and direction of the relationship in one number.
- The sign shows direction; the size shows strength: −0.82 is stronger than +0.60; +0.9 and −0.9 are equally strong.
- Scatter diagram: paired scores plotted as dots. A narrow band rising to the right = high positive; falling to the right = high negative; all on a rising straight line = +1; a shapeless cloud = zero.
- Pearson's r suits interval or ratio scores with a linear relationship; "product-moment" = the products of paired deviations from the two means. A strong curved (inverted-U) relationship can still give r near 0.
- Spearman's ρ uses ranks; quicker for small groups (N below about 25) or data available only as ranks. D = the difference between the two ranks of each pupil. Tied scores get the average of the ranks they share (three pupils sharing 2nd, 3rd and 4th each get 3).
- Spearman example: ranks (1, 2, 3, 4, 5) and (2, 1, 4, 3, 5); D: −1, 1, −1, 1, 0; ΣD² = 4. ρ = 1 − (6 × 4) ÷ (5 × 24) = 0.8. Identical ranks → ρ = +1; exactly reversed ranks → ρ = −1. N = 5, ΣD² = 10 → ρ = 0.5; N = 6, ΣD² = 14 → ρ = 0.6.
- r² (coefficient of determination): the share of variance the two have in common: r = 0.7 → 49%.
- Usual reading of r (Garrett): 0 to ±0.20 negligible; ±0.20 to ±0.40 low; ±0.40 to ±0.70 moderate (substantial); ±0.70 to ±0.90 high; ±0.90 to ±1.00 very high.
- Correlation is not causation: correlational studies ask whether two variables are associated; a significant r, positive or negative, only says the two vary together.
- Tests use correlation: test-retest reliability and validity coefficients are correlation coefficients (see EDU-ST-02).
14. Scales of measurement (S.S. Stevens, 1946; they decide which statistic to use)
| Scale | Example | Average |
|---|---|---|
| Nominal | gender, blood group, roll number | mode |
| Ordinal | ranks, grades A, B, C | median |
| Interval (no true zero) | test scores, temperature in °C | mean |
| Ratio (true zero) | height, weight, time | mean (and all others) |
Common traps
- ✗ "Median is the mathematical average." ✓ Mean is mathematical; median is positional.
- ✗ "Mean is right for nominal data." ✓ Mode is the only average for nominal data.
- ✗ "The score above and below which half the scores fall is the mean." ✓ It is the median.
- ✗ "SD is the square of the variance." ✓ SD is the square root of the variance; variance is the square of SD.
- ✗ "Variance is the average difference from the mean." ✓ It is the average of the squared differences.
- ✗ "Highest − lowest score is the quartile deviation." ✓ That is the range; Q = (Q₃ − Q₁) ÷ 2.
- ✗ "Norms, reliability or validity measure the spread of scores." ✓ Spread is measured by the SD (range, Q, MD); the others are qualities of a test.
- ✗ "A cumulative frequency graph is a histogram." ✓ It is an ogive.
- ✗ "Combined mean of 15 items (mean 8) and 10 items (mean 6) is 7." ✓ Weight by size: 180 ÷ 25 = 7.2.
- ✗ "Negative correlation means no relationship" or "−0.6 is weaker than +0.6". ✓ Negative = one rises as the other falls; −0.6 and +0.6 are equally strong; no relationship is zero correlation.
- ✗ "A significant correlation proves X causes Y." ✓ It only shows X and Y vary together.
- ✗ "Adding 5 grace marks to all raises the SD." ✓ The SD stays the same; only the averages rise by 5.
- ✗ BRAOU's BRS text calls ±0.40 to ±0.70 "high". ✓ The standard (Garrett) reading is moderate (substantial); high starts at ±0.70.
Quick revision
- Averages: mean (mathematical), median (positional, half above and half below), mode (most frequent).
- Nominal → mode; ranks, grades, skewed or open-ended → median; interval → mean.
- Median = ((N + 1) ÷ 2)th score after arranging; combined mean 15 × 8 and 10 × 6 → 7.2.
- Mode = 3 Median − 2 Mean; ogive = cumulative frequency graph.
- Range = H − L; Q = (Q₃ − Q₁) ÷ 2; MD ignores signs.
- Variance = mean of squared deviations; SD = √variance; SD 0 = all scores equal.
- Grace marks: averages rise, SD unchanged; CV = SD ÷ mean × 100.
- Normal curve 68.26 / 95.44 / 99.73% within 1 / 2 / 3 SD; z mean 0 SD 1; T mean 50 SD 10.
- Correlation −1 to +1: positive together, negative opposite, zero none; not causation.
- Pearson r (product-moment), Spearman ρ = 1 − 6ΣD² ÷ [N(N² − 1)], Galton first.
Go deeper: Deeper (beyond what APPSC Paper I usually asks)
Deeper Advanced
1. Standard error of the mean (SEM) = SD ÷ √N: how much the mean of a sample would vary from sample to sample. It is not the SD of the scores. SEM, parameter (a value for the population) and estimate are stock distractors in "variance" and "SD" questions.
2. Shape of a curve. Skewness = lack of symmetry; kurtosis = peakedness: leptokurtic (more peaked than normal), mesokurtic (normal), platykurtic (flatter).
3. Other correlation coefficients
| Coefficient | Used when |
|---|---|
| Phi (φ) | both variables are true two-category (gender, pass/fail) |
| Point-biserial | one two-category (item right/wrong), one continuous (total score): item discrimination |
| Partial | the effect of a third variable (age) is removed |
| Multiple (R) | one variable is related to two or more others together |
4. Cautions with r. Adding marks to one test does not change r. A restricted range (a class of toppers only) lowers r. With a large sample even a small r is statistically significant. Correlation coefficients are not percentages and cannot be averaged directly.
Practise this topic. Questions on Measures of central tendency, with answers explained.
All Statistics for teachers notes · APPSC Paper I syllabus · APPSC Paper I Mock 1 with answers · Start free
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