National 5 - Standard Deviation worksheet - Free Printable

Let's solve the problem step by step to find the standard deviation of the number set:
6, 2, 9, 3, 5, 5

---

Step 1: Find the mean ($\bar{x}$)

$$
\bar{x} = \frac{\text{Sum of all values}}{\text{Number of values}}
$$

$$
\bar{x} = \frac{6 + 2 + 9 + 3 + 5 + 5}{6} = \frac{30}{6} = 5
$$

So,
$$
\bar{x} = 5
$$

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Step 2: Complete the table

We'll fill in the table with:

- $x$: each data point
- $\bar{x}$: the mean (which is 5 for all rows)
- $x - \bar{x}$: difference from the mean
- $(x - \bar{x})^2$: square of the difference

| $x$ | $\bar{x}$ | $x - \bar{x}$ | $(x - \bar{x})^2$ |
|------|------------|------------------|----------------------|
| 6 | 5 | $6 - 5 = 1$ | $1^2 = 1$ |
| 2 | 5 | $2 - 5 = -3$ | $(-3)^2 = 9$ |
| 9 | 5 | $9 - 5 = 4$ | $4^2 = 16$ |
| 3 | 5 | $3 - 5 = -2$ | $(-2)^2 = 4$ |
| 5 | 5 | $5 - 5 = 0$ | $0^2 = 0$ |
| 5 | 5 | $5 - 5 = 0$ | $0^2 = 0$ |

Now, add up the last column:

$$
\sum(x - \bar{x})^2 = 1 + 9 + 16 + 4 + 0 + 0 = 30
$$

And total of $x - \bar{x}$ is:
$$
1 + (-3) + 4 + (-2) + 0 + 0 = 0 \quad \text{(as expected)}
$$

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Step 3: Use the standard deviation formula

$$
s.d = \sqrt{ \frac{\sum(x - \bar{x})^2}{n - 1} }
$$

Here:
- $\sum(x - \bar{x})^2 = 30$
- $n = 6$, so $n - 1 = 5$

$$
s.d = \sqrt{ \frac{30}{5} } = \sqrt{6} \approx 2.45
$$

---

Final Answer:

$$
\boxed{
\bar{x} = 5
}
$$

| $x$ | $\bar{x}$ | $x - \bar{x}$ | $(x - \bar{x})^2$ |
|------|------------|------------------|----------------------|
| 6 | 5 | 1 | 1 |
| 2 | 5 | -3 | 9 |
| 9 | 5 | 4 | 16 |
| 3 | 5 | -2 | 4 |
| 5 | 5 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| | | Totals | 30 |

$$
s.d = \sqrt{ \frac{30}{5} } = \sqrt{6} \approx 2.45
$$

✔ So, the standard deviation is $\boxed{\sqrt{6}}$ or approximately $\boxed{2.45}$.