📋 1 · Descriptive Statistics
Research Question:
A professor records exam scores for 10 students. Describe the distribution.
| Student | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|
| Score | 72 | 85 | 90 | 68 | 78 | 92 | 75 | 88 | 65 | 82 |
-
1
Mean (x̄):
$$\bar{x} = \frac{\sum x_i}{n} = \frac{72+85+90+68+78+92+75+88+65+82}{10} = \frac{795}{10} = 79.5$$
-
2
Sample Standard Deviation:
$$s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}} = \sqrt{\frac{(-7.5)^2+\cdots+(2.5)^2}{9}} = \sqrt{\frac{630.5}{9}} = \sqrt{70.06} \approx 8.37$$
-
3
Median (sorted: 65,68,72,75,78,82,85,88,90,92):
$$\text{Median} = \frac{x_{(5)} + x_{(6)}}{2} = \frac{78 + 82}{2} = 80.0$$
-
4
Standard Error of the Mean:
$$SE = \frac{s}{\sqrt{n}} = \frac{8.37}{\sqrt{10}} = \frac{8.37}{3.162} = 2.65$$
APA-7
Descriptive statistics for exam scores (N = 10): M = 79.50, SD = 8.37, Mdn = 80.00, range = [65, 92].