1. Variance

Formula (sample):

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

Practice Example:
Dataset:

4, 8, 6, 10

Steps:

1. Find the mean
   $\bar{x} = (4+8+6+10) / 4 = 28/4 = 7$

2. Subtract the mean from each number and square the result

   • (4−7)²=9

   • (8−7)²=1

   • (6−7)²=1

   • (10−7)²=9

3. Add squared differences = 20

4. Divide by $n−1 = 3$:

$$s^2 = 20/3 ≈ 6.67$$

✔ Try:
5, 7, 12, 15, 18 — find variance step by step.

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 2. Standard Deviation (SD)

Formula:

$$s = \sqrt{s^2}$$

Continue the example above:

Variance ≈ 6.67
Standard Deviation:

$$s = \sqrt{6.67} ≈ 2.58$$

Practice Tasks:

• Use the previous dataset and complete the SD calculation.

• Practice with:

  3, 5, 7, 9, 11
  

Tip: Always follow same steps — mean → deviations → squared → average → square root.

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 3. Coefficient of Variation (CV)

Formula:

$$CV = \left(\frac{SD}{Mean}\right) \times 100\%$$

Practice Example:
Dataset:

10, 12, 14, 16, 18

1. Mean:

$$\bar{x} = (10+12+14+16+18)/5 = 70/5 = 14$$

2. Find SD with steps above (you can calculate it or use a calculator).

3. Suppose SD = 2.83
   Then,

$$CV = (2.83/14) × 100 ≈ 20.2\%$$

👉 A higher CV means more variation relative to the mean; a lower CV means data is more consistent relative to its average.

Practice CV Problems:

1. Dataset: 50, 60, 70, 80, 90 → find CV.

2. Dataset: 2, 4, 6, 8, 10 → find SD and CV.

3. Two datasets:

   • A: 15, 18, 20, 22, 25

   • B: 150, 180, 200, 220, 250
     → Which has higher relative variability?

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 Practice Method You Can Repeat

For each dataset:

1. Find mean

2. Compute square deviations from the mean

3. Find variance

4. Take square root → SD

5. Use SD and mean to compute CV

Start with small data (5–8 numbers), then try larger ones.

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 Why Practice This Way

Practicing step by step reinforces:

• How each measure builds on the previous (mean → variance → SD → CV)

• Understanding of how spread relates to the mean

• Ability to compare variability across datasets — especially using CV