Let us compute the standard deviation of the height of nine students that we used while calculating Mean. The Mean (x̄) was calculated to be 101.33 cm. Subtract each value from the mean and take square of that value. Dividing the sum of square values by total number of values and

Question

Let us compute the standard deviation of the height of nine students that we used while calculating Mean. The Mean (x̄) was calculated to be 101.33 cm. Subtract each value from the mean and take square of that value. Dividing the sum of square values by total number of values and taking its square root gives the standard deviation in data. See Table 7.3 for details.

Accepted Answer

> [!TLDR]
> Standard deviation measures how spread out the data is around the mean. For the nine heights used throughout this section (Table 7.3: 90, 102, 110, 115, 85, 90, 100, 110, 110 cm, mean 101.33 cm), $\sigma = \sqrt{938.00 / 9} = \sqrt{104.22} \approx$ **10.2 cm**.

This is a **plain theory / calculation question** — you are being asked to complete the standard deviation calculation for the same nine heights used in Examples 7.1–7.4, using the mean that has already been found (101.33 cm) and Table 7.3.

## The Idea: Why Standard Deviation?

The arithmetic mean tells you the "centre" of the data, but it says nothing about how the individual values are scattered around that centre. Standard deviation solves this by measuring the **average distance** of each data point from the mean. Squaring each deviation removes the cancellation that would occur if positive and negative deviations were simply averaged, and taking the square root at the end brings the units back to the original scale (cm, not cm²).

$$ \sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}} $$

## The Calculation (Table 7.3)

The nine heights (in cm), the same ones used to compute the mean of 101.33 cm, are: 90, 102, 110, 115, 85, 90, 100, 110, 110.

| Height ($x$) | $x - \bar{x}$ | $(x - \bar{x})^2$ |
|---|---|---|
| 90  | $-11.33$ | $128.37$ |
| 102 | $0.67$   | $0.36$   |
| 110 | $8.67$   | $75.17$  |
| 115 | $13.67$  | $186.87$ |
| 85  | $-16.33$ | $266.67$ |
| 90  | $-11.33$ | $128.37$ |
| 100 | $-1.33$  | $1.77$   |
| 110 | $8.67$   | $75.17$  |
| 110 | $8.67$   | $75.17$  |

**Step 1:** Subtract the mean from each height, and square the result — shown in the middle and right columns above.

**Step 2:** Sum the squared deviations: $\sum (x - \bar{x})^2 = 938.00$.

**Step 3:** Divide by the number of values ($n = 9$): $938.00 / 9 = 104.22$ — this is the variance.

**Step 4:** Take the square root: $\sigma = \sqrt{104.22} \approx 10.2$ cm.

```python
heights = [90, 102, 110, 115, 85, 9

X	f	fX
1	5	5
2	9	18
3	12	36
4	17	68
5	14	70
6	10	60
7	6	42
Total	73	299

Marks (X)	1	2	3	4	5
f	8	10	Y	2	4
fX	8	20	3Y	8	20

Height ( $x$ )	Deviation ( $x - \bar{x}$ )	Squared deviation $(x - \bar{x})^2$
90	$-11.33$	$128.37$
102	$0.67$	$0.36$
110	$8.67$	$75.17$
115	$13.67$	$186.87$
85	$-16.33$	$266.67$
90	$-11.33$	$128.37$
100	$-1.33$	$1.77$
110	$8.67$	$75.17$
110	$8.67$	$75.17$

The Idea: Why Standard Deviation?

The Calculation (Table 7.3)

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