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Statistics in education

Teacher Education / Education Studies

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OSSC Science, SSB Lecturer (Degree Colleges)

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Overview

Averages describe the centre of data and measures of dispersion describe its spread; grouped-data formulas are central to .

Key points

For positive data, AM ≥ GM ≥ HM; for two numbers, GM² = AM × HM.
Grouped median = L + ((N/2 − cf)/f) × h.
Grouped mode = L + ((f₁ − f₀)/(2f₁ − f₀ − f₂)) × h.
Quartile deviation = (Q₃ − Q₁)/2; coefficient of QD = (Q₃ − Q₁)/(Q₃ + Q₁).
Coefficient of variation = (SD/mean) × 100%.

Explanation

Mean deviation is least when taken about the median.
SD is affected by change of scale but not by change of origin.
Combined SD needs group means, SDs and sizes.
Lower CV means greater consistency.

Step by step

Identify the measure or test asked and its formula.
Organise the data (table, totals) before computing.
Interpret the result (sign, size, significance).

Examples

For 4 and 9: AM = 6.5, GM = 6, HM = 2 × 4 × 9/13 = 72/13 ≈ 5.54; check GM² = 36 = 6.5 × 72/13.
Q₁ = 20, Q₃ = 40 → QD = 10; coefficient = 20/60 = 0.33.
Series A: mean 50, SD 5 (CV 10%); series B: mean 40, SD 6 (CV 15%) → A is more consistent.

Traps and advanced points

Trap: Using N instead of N/2 in the median formula.
Trap: Thinking SD changes when a constant is added.
Trap: Comparing SDs instead of CVs for consistency.

Quick revision

AM ≥ GM ≥ HM.
Median and mode grouped formulas.
CV for consistency.

Practice MCQs

The mean of 2, 4 and 6 is —
1. 3
2. 4
3. 6
4. 12
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Answer: (b) 4. (2 + 4 + 6) ÷ 3 = 4.
The value that occurs most often in data is the —
1. mean
2. median
3. mode
4. range
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Answer: (c) mode. Mode = most frequent value.
The middle value of data arranged in order is the —
1. mean
2. median
3. mode
4. deviation
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Answer: (b) median. Median divides ordered data into two equal halves.
The coefficient of correlation lies between —
1. 0 and 1
2. −1 and +1
3. 0 and 100
4. −10 and +10
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Answer: (b) −1 and +1. +1 perfect positive, −1 perfect negative, 0 none.
Mean = 20 and SD = 5. The z-score of a score of 30 is —
1. 1
2. 1.5
3. 2
4. 6
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Answer: (c) 2. z = (30 − 20) ÷ 5 = 2.
Which measure is most affected by extreme values?
1. mean
2. median
3. mode
4. quartile
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Answer: (a) mean. The mean uses every value, so outliers pull it.
Which of these is a non-parametric test?
1. t-test
2. z-test
3. chi-square test
4. ANOVA
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Answer: (c) chi-square test. Chi-square works on frequencies without assuming normality.
In a normal distribution, about what percentage of scores lie within ±1 SD of the mean?
1. 50%
2. 68%
3. 95%
4. 99.7%
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Answer: (b) 68%. 68–95–99.7 rule: ±1 SD ≈ 68%.
For positive data, the correct order is —
1. HM ≥ GM ≥ AM
2. GM ≥ AM ≥ HM
3. all equal always
4. AM ≥ GM ≥ HM
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Answer: (d) AM ≥ GM ≥ HM. Mean inequality.
For two positive numbers, GM² equals —
1. AM + HM
2. AM ÷ HM
3. AM − HM
4. AM × HM
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Answer: (d) AM × HM. Relation of means.