Question

Which data set is the most spread from its mean?
14, 26, 24, 28
22, 16, 18, 36
22, 28, 20, 22
21, 19, 27, 25

Answer 1

To determine which data set is the most spread from its mean, we can calculate the variance and standard deviation for each data set.

1. Calculate the mean for each data set.
2. Calculate the variance (the average of the squared differences from the mean).
3. The data set with the highest variance will be the most spread out.

Let's calculate this step-by-step for each data set.

### Data Set 1: 14, 26, 24, 28
- Mean: \( \text{Mean} = \frac{14 + 26 + 24 + 28}{4} = \frac{92}{4} = 23 \)
- Variance:
\[
\text{Variance} = \frac{(14-23)^2 + (26-23)^2 + (24-23)^2 + (28-23)^2}{4} = \frac{(-9)^2 + 3^2 + 1^2 + 5^2}{4} = \frac{81 + 9 + 1 + 25}{4} = \frac{116}{4} = 29
\]

### Data Set 2: 22, 16, 18, 36
- Mean: \( \text{Mean} = \frac{22 + 16 + 18 + 36}{4} = \frac{92}{4} = 23 \)
- Variance:
\[
\text{Variance} = \frac{(22-23)^2 + (16-23)^2 + (18-23)^2 + (36-23)^2}{4} = \frac{(-1)^2 + (-7)^2 + (-5)^2 + 13^2}{4} = \frac{1 + 49 + 25 + 169}{4} = \frac{244}{4} = 61
\]

### Data Set 3: 22, 28, 20, 22
- Mean: \( \text{Mean} = \frac{22 + 28 + 20 + 22}{4} = \frac{92}{4} = 23 \)
- Variance:
\[
\text{Variance} = \frac{(22-23)^2 + (28-23)^2 + (20-23)^2 + (22-23)^2}{4} = \frac{(-1)^2 + 5^2 + (-3)^2 + (-1)^2}{4} = \frac{1 + 25 + 9 + 1}{4} = \frac{36}{4} = 9
\]

### Data Set 4: 21, 19, 27, 25
- Mean: \( \text{Mean} = \frac{21 + 19 + 27 + 25}{4} = \frac{92}{4} = 23 \)
- Variance:
\[
\text{Variance} = \frac{(21-23)^2 + (19-23)^2 + (27-23)^2 + (25-23)^2}{4} = \frac{(-2)^2 + (-4)^2 + 4^2 + 2^2}{4} = \frac{4 + 16 + 16 + 4}{4} = \frac{40}{4} = 10
\]

### Summary of Variances
- Data Set 1: Variance = 29
- Data Set 2: Variance = 61
- Data Set 3: Variance = 9
- Data Set 4: Variance = 10

### Conclusion
The data set that is the most spread from its mean is **Data Set 2: 22, 16, 18, 36**, with a variance of 61.