Words you need first
- Data: a list of numbers we collected, like test marks.
- Mean (x̄): add all values, divide by how many. It is the balance point.
- Median (M): the middle value after arranging in order.
- Deviation: how far one value is from the centre, x − x̄. It can be minus or plus.
- Dispersion: spread. A measure of dispersion is one number that tells how spread out the data is.
Why do we need it? Two lists can have the same mean but look very different. The mean alone cannot tell them apart.
Range
Range = largest value − smallest value.
It is quick. But it looks at only two numbers. One very big or very small value (an outlier) can make the range huge even if all other values are close together. So we need better measures.
Mean deviation for ungrouped data
Find the distance of each value from a centre a (the mean or the median). Distance means we ignore the minus sign: |x − a|.
M.D.(a) = Σ|xᵢ − a| ÷ n
Steps: (1) find the mean or median, (2) write each |x − a|, (3) add them, (4) divide by n.
Why the absolute value? The plain deviations x − x̄ always add up to 0. Minus and plus cancel. Removing signs stops this.
Fact: mean deviation about the median is the smallest possible mean deviation.
Mean deviation for grouped data
Discrete frequency table (values x with frequencies f, N = Σf): M.D.(a) = Σfᵢ|xᵢ − a| ÷ N.
Continuous classes (like 0–10, 10–20): use each class mid-value as x, then the same formula.
For the median of a continuous table: find N/2, locate the median class with the cumulative frequency, then Median = l + ((N/2 − cf) ÷ f) × h, where l is the lower limit, cf the cumulative frequency before the class, f its frequency and h the class width.
Variance and standard deviation for ungrouped data
Variance σ² = Σ(xᵢ − x̄)² ÷ n. Squaring removes minus signs and gives big distances extra weight.
Standard deviation σ = √variance. It is back in the same unit as the data (marks, cm, runs).
Shortcut formula (useful when the mean is a decimal): σ² = Σx² ÷ n − (x̄)², or σ = (1/n)√(nΣx² − (Σx)²).
σ is never negative. σ = 0 only when every value is the same.
Variance and standard deviation for grouped data
Discrete or continuous (use mid-values x): σ² = Σfᵢ(xᵢ − x̄)² ÷ N, or the shortcut σ² = Σfx² ÷ N − (x̄)².
Step-deviation method (saves work with big numbers): pick an assumed mean A and class width h, set u = (x − A) ÷ h. Then
σ² = h² [ Σfu² ÷ N − (Σfu ÷ N)² ] and x̄ = A + h × Σfu ÷ N.
Changing origin (adding a number to every value) does not change σ. Multiplying every value by k multiplies σ by |k|.
Comparing two data sets: coefficient of variation
When the means are different, compare spread with C.V. = (σ ÷ x̄) × 100. The set with the smaller C.V. is more consistent (more stable).
Try it: measure your family's spread
Ask 5 people at home how many minutes they slept last night (or use 5 test marks). Write the numbers. Find the range, then the mean, then each distance from the mean, then the squares. Work out σ. Now type the same 5 numbers with the sliders on the last 3D step and check your answer. Then move one slider far away and see how much σ jumps.
Board exam pattern
The Statistics and Probability unit carries 12 marks in CBSE Class 11. A typical long question asks for the mean deviation or the variance and SD of a grouped table. Always show the table with columns x, f, fx, d or u, fd², so step marks are safe.
Key formulas and definitions
- Range = largest value − smallest value
- M.D.(a) = Σ|xᵢ − a| ÷ n (a = mean or median)
- Grouped: M.D.(a) = Σfᵢ|xᵢ − a| ÷ N
- Variance σ² = Σ(xᵢ − x̄)² ÷ n = Σx² ÷ n − x̄²
- Grouped: σ² = Σfᵢ(xᵢ − x̄)² ÷ N
- Step deviation: σ² = h²[Σfu² ÷ N − (Σfu ÷ N)²], u = (x − A) ÷ h
- Standard deviation σ = √variance
- C.V. = (σ ÷ x̄) × 100
Worked examples
1. Find the range of 12, 7, 19, 3, 15.
Largest = 19, smallest = 3. Range = 19 − 3 = 16.
2. Find the mean deviation about the mean for 3, 5, 7, 9, 11.
Mean = 35 ÷ 5 = 7. Distances |x − 7|: 4, 2, 0, 2, 4. Sum = 12. M.D. = 12 ÷ 5 = 2.4.
3. Find the mean deviation about the median for 4, 7, 8, 9, 10, 12, 13, 17.
8 values in order, so median = (9 + 10) ÷ 2 = 9.5. Distances: 5.5, 2.5, 1.5, 0.5, 0.5, 2.5, 3.5, 7.5. Sum = 24. M.D. = 24 ÷ 8 = 3.
4. Find the variance and standard deviation of 6, 8, 10, 12, 14.
Mean = 50 ÷ 5 = 10. Deviations: −4, −2, 0, 2, 4. Squares: 16, 4, 0, 4, 16. Sum = 40. Variance = 40 ÷ 5 = 8. SD = √8 ≈ 2.83.
5. For the table x: 2, 4, 6, 8 with f: 1, 3, 4, 2, find the mean deviation about the mean and the standard deviation.
N = 10. Σfx = 2 + 12 + 24 + 16 = 54, so x̄ = 5.4. |x − 5.4|: 3.4, 1.4, 0.6, 2.6. f|x − 5.4|: 3.4, 4.2, 2.4, 5.2, sum 15.2, so M.D. = 1.52. Σfx² = 4 + 48 + 144 + 128 = 324. σ² = 324 ÷ 10 − 5.4² = 32.4 − 29.16 = 3.24. σ = 1.8.
6. Classes 0–10, 10–20, 20–30, 30–40, 40–50 have frequencies 2, 3, 5, 3, 2. Find the mean deviation about the mean, the variance and the SD.
Mid-values x: 5, 15, 25, 35, 45; N = 15. Σfx = 10 + 45 + 125 + 105 + 90 = 375, x̄ = 25. |x − 25|: 20, 10, 0, 10, 20; f × that: 40, 30, 0, 30, 40 = 140; M.D. = 140 ÷ 15 ≈ 9.33. f(x − 25)²: 800, 300, 0, 300, 800 = 2200. σ² = 2200 ÷ 15 ≈ 146.67. σ ≈ 12.11.
7. Do the same table (2, 3, 5, 3, 2 over 0–50) by the step-deviation method.
Take A = 25, h = 10, u = (x − 25) ÷ 10: −2, −1, 0, 1, 2. fu: −4, −3, 0, 3, 4, Σfu = 0. fu²: 8, 3, 0, 3, 8, Σfu² = 22. σ² = 10² × [22 ÷ 15 − 0²] = 100 × 1.4667 ≈ 146.67. Same answer with smaller numbers.
8. Classes 0–10, 10–20, 20–30, 30–40 have frequencies 4, 6, 8, 2. Find the mean deviation about the median.
N = 20, N/2 = 10. Cumulative f: 4, 10, 18, 20. The first class whose cf is more than 10 is 20–30, so l = 20, cf = 10, f = 8, h = 10. Median = 20 + ((10 − 10) ÷ 8) × 10 = 20. Mid-values 5, 15, 25, 35; |x − 20|: 15, 5, 5, 15; f × that: 60, 30, 40, 30 = 160. M.D. = 160 ÷ 20 = 8.
9. Batter A: mean 50, SD 10. Batter B: mean 80, SD 12. Who is more consistent?
C.V.(A) = 10 ÷ 50 × 100 = 20%. C.V.(B) = 12 ÷ 80 × 100 = 15%. B has the smaller C.V., so B is more consistent.
Common mistakes
- Adding plain deviations (x − x̄). They always add to 0. Use |x − x̄| for mean deviation or (x − x̄)² for variance.
- Forgetting to multiply by f in grouped data, or dividing by the number of classes instead of N = Σf.
- Reporting the variance when the question asks for standard deviation. Take the square root at the end.
- In the step-deviation method, forgetting to multiply by h² at the end.