The essence of this chapter for last-minute revision: the three techniques of data analysis, the definitions of mean/median/mode, every formula for ungrouped and grouped data, all six worked examples from the chapter with their answers, and how the three measures relate in normal and skewed distributions. The full notes, with the complete explanation, are available separately.
1 Definitions to learn by heart
Three techniques analyse data: Measures of Central Tendency, Measures of Dispersion and Measures of Relationship. This chapter covers only the first.
Measures of central tendency (also called statistical averages) give a single representative value lying near the centre of a distribution: the mean, the median and the mode.
The mean (symbol X̄) is the value derived by summing all values and dividing by the number of observations (N). The median (symbol M) is a positional average, the point with an equal number of cases on each side. The mode (symbol Z or M0) is the value occurring with the maximum frequency.
2 Mean: all four formulas and worked examples
| Method | Formula | Worked example | Answer |
|---|---|---|---|
| Ungrouped, direct | X̄ = Σx / N | Ex. 2.1: Malwa Plateau rainfall (7 districts: Indore 979, Dewas 1,083, Dhar 833, Ratlam 896, Ujjain 891, Mandsaur 825, Shajapur 977 mm); Σx=6,484, N=7 | 926.29 mm |
| Ungrouped, indirect | X̄ = A + Σd/N (d=x−A) | Same data, A=800 (assumed mean); Σd=884 | 926.29 mm (same answer) |
| Grouped, direct | X̄ = Σfx / N | Ex. 2.2: factory wage rates, classes 50-70/70-90/90-110/110-130/130-150 with f=10,20,25,35,9; Σfx=10,160, N=99 | ₹102.6/day |
| Grouped, indirect | X̄ = A ± Σfd/N | Same data, assumed mean class 90-110 (A=100); Σfd=260 | ₹102.6/day (same answer) |
The indirect method is normally used for a large number of observations: it reduces values to smaller numbers (“coding”) by subtracting an assumed mean (A), chosen as close as possible to the middle of the series to minimise computation. It works for both equal and unequal class intervals, and always gives the same answer as the direct method.
3 Median: both formulas and worked examples
| Data type | Formula | Worked example | Answer |
|---|---|---|---|
| Ungrouped | Value of the (N+1)/2th item (data arranged in order) | Ex. 2.3: Himalayan peak heights 8,126; 8,611; 7,817; 8,172; 8,076; 8,848; 8,598 m (N=7); 4th item of the arranged series | 8,172 m |
| Grouped | M = l + (i/f)(N/2 − c) | Ex. 2.4: classes 50-60 to 100-110, f=3,7,11,16,8,5 (N=50); median class 80-90 (l=80, f=16, c=21, i=10) | 82.5 |
l = lower limit of the median class · i = class interval · f = frequency of the median class · c = cumulative frequency of the class just before the median class. The median class is the one whose cumulative frequency first exceeds N/2.
4 Mode: worked examples and the five categories
Ex. 2.5: geography test scores 61, 10, 88, 37, 61, 72, 55, 61, 46, 22 → 61 repeats three times → mode = 61 (unimodal).
Ex. 2.6: scores 82, 11, 57, 82, 08, 11, 82, 95, 41, 11 → both 11 and 82 repeat three times → modes = 11 and 82 (bimodal).
| Category | Meaning |
|---|---|
| Unimodal | Only one value has the highest frequency |
| Bimodal | Two values share an equal, highest frequency |
| Trimodal | Three values share an equal, highest frequency |
| Multimodal | Many values are repeated |
| Without mode | No value is repeated at all |
5 Comparison of mean, median and mode
| Distribution | Order of the three measures |
|---|---|
| Normal (symmetrical, Fig. 2.1) | Mean = Median = Mode (all coincide, e.g. all = 100 in the chapter’s own figure) |
| Positive skew (Fig. 2.2) | Mode < Median < Mean (long tail towards higher values pulls the mean right) |
| Negative skew (Fig. 2.3) | Mean < Median < Mode (long tail towards lower values pulls the mean left) |
In a normal distribution, the highest-frequency score sits exactly in the middle, with half the observations above and half below. In any skew, the mode shifts the least (it just follows the peak) and the mean shifts the most (it is computed from every value, extremes included).
- Three techniques: central tendency, dispersion, relationship (this chapter = central tendency only)
- Mean X̄=Σx/N (ungrouped direct) · X̄=A+Σd/N (ungrouped indirect) · X̄=Σfx/N (grouped direct) · X̄=A±Σfd/N (grouped indirect)
- Median (ungrouped) = value of (N+1)/2th item · Median (grouped) M=l+(i/f)(N/2−c)
- Ex. 2.1 rainfall mean = 926.29 mm · Ex. 2.2 wage mean = ₹102.6/day · Ex. 2.3 peak-height median = 8,172 m · Ex. 2.4 grouped median = 82.5
- Ex. 2.5 mode = 61 (unimodal) · Ex. 2.6 modes = 11 & 82 (bimodal)
- Normal: Mean=Median=Mode · Positive skew: Mode<Median<Mean · Negative skew: Mean<Median<Mode