Formula and Uses in Statistics with a small simple example.

1. Mean (raw scores)

Formula:

$$\bar{X} = \frac{\Sigma X}{N}$$

Example:
Scores = 10, 15, 20, 25
$\Sigma X = 10 + 15 + 20 + 25 = 70$
$N = 4$

$$\bar{X} = \frac{70}{4} = 17.5$$

✅ Mean = 17.5

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2. Mean (grouped data)

Formula:

$$\bar{X} = \frac{\Sigma fX}{\Sigma f}$$

Example:

Class	Midpoint (X)	f	fX
0–9	5	2	10
10–19	15	3	45
20–29	25	5	125

$\Sigma f = 2 + 3 + 5 = 10$
$\Sigma fX = 10 + 45 + 125 = 180$

$$\bar{X} = \frac{180}{10} = 18$$

✅ Mean = 18

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3. Standard Deviation (SD, grouped data)

Formula:

$$SD = \sqrt{\frac{\Sigma fX^2}{\Sigma f} - \left(\frac{\Sigma fX}{\Sigma f}\right)^2}$$

Example (same table):

X	f	fX	fX²
5	2	10	50
15	3	45	675
25	5	125	3125

$\Sigma f = 10$, $\Sigma fX = 180$, $\Sigma fX² = 3850$.

$$SD = \sqrt{\frac{3850}{10} - \left(\frac{180}{10}\right)^2}$$ $$= \sqrt{385 - 18^2} = \sqrt{385 - 324} = \sqrt{61} ≈ 7.8$$

✅ SD ≈ 7.8

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4. Z–score

Formula:

$$Z = \frac{X - \bar{X}}{SD}$$

Example:
Mean ($\bar{X}$) = 50, SD = 10, Raw score $X = 70$.

$$Z = \frac{70 - 50}{10} = \frac{20}{10} = 2$$

✅ Z = 2

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5. Raw score from Z

Formula:

$$X = Z \times SD + \bar{X}$$

Example:
Z = –1.5, Mean = 50, SD = 10.

$$X = -1.5 \times 10 + 50 = -15 + 50 = 35$$

✅ Raw score = 35

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6. T–score

Formula:

$$T = 50 + 10Z$$

Example:
Z = 2 → $T = 50 + 10(2) = 70$.
Z = –1.5 → $T = 50 + 10(-1.5) = 35$.

✅ T = 70 and 35

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7. Degree of freedom (df)

Formula:

$$df = N - 1$$

Example:
If you have 6 values,

$$df = 6 - 1 = 5$$

✅ df = 5