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قدم ب قدم

1. 10, 12, 11, 13

   4 values.

2. \bar{x} = \frac{10 + 12 + 11 + 13}{4} = \frac{46}{4} = \frac{23}{2}

   Mean: add them up and divide by how many there are.

3. (10 - \frac{23}{2})^2 = \frac{9}{4} ,\; (12 - \frac{23}{2})^2 = \frac{1}{4} ,\; (11 - \frac{23}{2})^2 = \frac{1}{4} ,\; (13 - \frac{23}{2})^2 = \frac{9}{4}

   Squared deviations from the mean.

4. \sigma^2 = \frac{\sum (x_i - \bar x)^2}{n} = \frac{5}{4}

   Population variance (divide by n).

5. s^2 = \frac{\sum (x_i - \bar x)^2}{n - 1} = \frac{5}{3}

   Sample variance (divide by n − 1).

6. \sigma = \sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2} \approx 1.1180,\quad s = \frac{\sqrt{15}}{3} \approx 1.2910

   Standard deviation is the square root of the variance.