Measures of Central Tendency & Dispersion
For BITSAT – Mathematics
Goal – Understand how to describe a data set by a single “typical” value (central tendency) and by how spread out the values are (dispersion).
1. Simple Explanation
| Concept | What it tells you | Why it matters |
|---|---|---|
| Mean | Average value (sum / number of observations). | Gives the “balance point” of the data. Sensitive to out‑liers. |
| Median | Middle value when data are ordered. | Robust against extreme values. |
| Mode | Most frequently occurring value(s). | Indicates the most common outcome(s). |
| Range | Max – Min. | Rough sense of spread; very sensitive to extremes. |
| Variance | Average of squared deviations from the mean. | Measures spread in the same units squared. |
| Standard Deviation (SD) | Square‑root of variance. | Gives spread in the same units as the data. |
| Mean Deviation (MD) | Average of absolute deviations from the mean. | Less affected by extreme values than variance. |
| Coefficient of Variation (CV) | SD / Mean (often ×100%). | Gives relative spread, useful when comparing data with different units. |
| Inter‑Quartile Range (IQR) | Q3 – Q1. | Spread of middle 50 % of data; robust to out‑liers. |
2. Key Points
| Measure | Formula | Interpretation | Special Notes |
|---|---|---|---|
| Mean (μ) | (\displaystyle \bar{x} = \frac{\sum x_i}{n}) | Central value; used in BITSAT when data are symmetric. | Out‑liers inflate the mean. |
| Median | Middle term of ordered list (or average of two middle terms). | Robust to extremes. | For even (n), median = (n/2)th + (n/2+1)th term / 2. |
| Mode | Value(s) occurring most often. | Indicates most frequent observation; may be multiple. | In continuous data, use histogram mode. |
| Range | (\max(x_i) - \min(x_i)) | Quick sense of spread. | Sensitive to single extreme values. |
| Variance (σ²) | (\displaystyle \frac{\sum (x_i - \bar{x})^2}{n}) (population) or (\displaystyle \frac{\sum (x_i - \bar{x})^2}{n-1}) (sample). | Measures average squared deviation. | Use (n-1) for sample to avoid under‑estimation. |
| Standard Deviation (σ) | (\sqrt{\text{Variance}}) | Spread in same units as data. | Often used in BITSAT questions. |
| Mean Deviation (MD) | (\displaystyle \frac{\sum | x_i - \bar{x} | }{n}) |
| Coefficient of Variation (CV) | (\displaystyle \frac{\sigma}{\bar{x}}\times 100%) | Relative spread; useful when comparing different datasets. | CV > 100 % means SD exceeds mean. |
| Inter‑Quartile Range (IQR) | (Q_3 - Q_1) | Spread of middle 50 %. | Robust to out‑liers; used in box‑plot. |
Quick Memorisation Trick
- Central Tendency → Mean, Median, Mode (order of sensitivity to out‑liers: Mode ≤ Median ≤ Mean)
- Dispersion → Range (most extreme) → IQR (middle 50 %) → MD → Variance → SD (square‑root gives back units)
3. Worked Example
Problem (BITSAT style)
A class of 10 students has the following test scores (out of 100):
72, 85, 78, 95, 88, 63, 90, 77, 84, 91
- Find the mean, median, and mode.
- Compute the range, variance, and standard deviation.
- Calculate the coefficient of variation.
- Determine the IQR.
Step‑by‑Step Solution
1. Central Tendency
| Statistic | Computation |
|---|---|
| Mean | (\displaystyle \bar{x}=\frac{72+85+78+95+88+63+90+77+84+91}{10}) (= \frac{813}{10}=81.3) |
| Median | Order the data: 63, 72, 77, 78, 84, 85, 88, 90, 91, 95. Middle two: 84 and 85 → Median = ((84+85)/2 = 84.5). |
| Mode | No score repeats → No mode. |
2. Dispersion
| Statistic | Formula | Result |
|---|---|---|
| Range | (\max - \min = 95-63 = 32). | |
| Variance | Population variance (since we have entire class): (\displaystyle \sigma^2 = \frac{\sum (x_i-\bar{x})^2}{n}). Compute deviations: |
|
| ((72-81.3)^2 = 84.49) | ||
| ((85-81.3)^2 = 13.69) | ||
| ((78-81.3)^2 = 10.89) | ||
| ((95-81.3)^2 = 187.69) | ||
| ((88-81.3)^2 = 44.89) | ||
| ((63-81.3)^2 = 335.29) | ||
| ((90-81.3)^2 = 76.89) | ||
| ((77-81.3)^2 = 18.49) | ||
| ((84-81.3)^2 = 7. |