Descriptive Statistics

Step-by-step solution

1. 1

   The data

   7 values, sorted in ascending order.

   $$4, 5, 6, 8, 8, 8, 9$$
2. 2

   Sum and mean

   Add all values, then divide the sum by n = 7. Decimal approximation: 48/7 ≈ 6.857143.

   $$\begin{aligned} \sum_{i=1}^{7} x_i &= 4 + 5 + 6 + 8 + 8 + 8 + 9 = 48 \\ \mu &= \frac{48}{7} = \frac{48}{7} \end{aligned}$$
3. 3

   Median

   With an odd number of values, the median is the middle value — position 4 of 7.

   $$\tilde{x} = 8$$
4. 4

   Mode

   The value 8 appears 3 times — more often than any other value, so it is the mode.

   $$\text{mode} = 8$$

   • 8 appears 3 times
5. 5

   Range

   The range is the difference between the largest and the smallest value.

   $$R = x_{\max} - x_{\min} = 9 - 4 = 5$$
6. 6

   Population variance σ²

   Square each deviation from the mean, add the squared deviations, and divide by n = 7.

   $$\sigma^2 = \frac{\sum_{i=1}^{n}(x_i - \mu)^2}{n} = \frac{\frac{146}{7}}{7} = \frac{146}{49}$$

   • (4 − 48/7)² = 400/49
   • (5 − 48/7)² = 169/49
   • (6 − 48/7)² = 36/49
   • (8 − 48/7)² = 64/49
   • (8 − 48/7)² = 64/49
   • (8 − 48/7)² = 64/49
   • (9 − 48/7)² = 225/49
7. 7

   Sample variance s²

   For a sample of a larger population, dividing by n − 1 = 6 instead of n (Bessel's correction) gives an unbiased estimate of the population variance.

   $$s^2 = \frac{\sum_{i=1}^{n}(x_i - \mu)^2}{n - 1} = \frac{\frac{146}{7}}{6} = \frac{73}{21}$$

   • (4 − 48/7)² = 400/49
   • (5 − 48/7)² = 169/49
   • (6 − 48/7)² = 36/49
   • (8 − 48/7)² = 64/49
   • (8 − 48/7)² = 64/49
   • (8 − 48/7)² = 64/49
   • (9 − 48/7)² = 225/49
8. 8

   Standard deviations

   The standard deviation is the square root of the variance. It is irrational in general, so only a decimal approximation is shown.

   $$\begin{aligned} \sigma = \sqrt{\sigma^2} = \sqrt{\frac{146}{49}} \approx 1.726149 \\ s = \sqrt{s^2} = \sqrt{\frac{73}{21}} \approx 1.864454 \end{aligned}$$
9. 9

   Quartiles

   Q1 is the median of the lower half (first 3 sorted values) and Q3 the median of the upper half (last 3 sorted values) — the middle value belongs to neither half.

   $$Q_1 = 5, \qquad Q_3 = 8$$
10. 10

    Result

    Mean 48/7 ≈ 6.857143, median 8, mode 8, population standard deviation σ ≈ 1.726149.

    $$\mu = \frac{48}{7}, \quad \tilde{x} = 8, \quad \sigma^2 = \frac{146}{49}, \quad \text{mode} = 8$$