Question

Which data set is the most spread from its mean?
26, 6, 20, 36
28, 15, 30, 15
49, 17, 4, 18
14, 20, 24, 30

Answer 1

To determine which dataset is the most spread from its mean, we can calculate the standard deviation for each dataset. The standard deviation provides a measure of the dispersion of a set of values around the mean.

### Step 1: Calculate the mean for each dataset.

1. **Dataset 1: 26, 6, 20, 36**
- Mean = (26 + 6 + 20 + 36) / 4 = 22.0

2. **Dataset 2: 28, 15, 30, 15**
- Mean = (28 + 15 + 30 + 15) / 4 = 22.0

3. **Dataset 3: 49, 17, 4, 18**
- Mean = (49 + 17 + 4 + 18) / 4 = 22.0

4. **Dataset 4: 14, 20, 24, 30**
- Mean = (14 + 20 + 24 + 30) / 4 = 19.5

### Step 2: Calculate the standard deviation for each dataset.

The formula for standard deviation (SD) is:
\[
SD = \sqrt{\frac{\sum (x_i - \text{mean})^2}{N}}
\]
where \(N\) is the number of data points.

1. **Dataset 1: 26, 6, 20, 36**
- Deviations: (26 - 22)^2 = 16, (6 - 22)^2 = 256, (20 - 22)^2 = 4, (36 - 22)^2 = 196
- Sum of squared deviations = 16 + 256 + 4 + 196 = 472
- SD = \(\sqrt{\frac{472}{4}} = \sqrt{118} \approx 10.86\)

2. **Dataset 2: 28, 15, 30, 15**
- Deviations: (28 - 22)^2 = 36, (15 - 22)^2 = 49, (30 - 22)^2 = 64, (15 - 22)^2 = 49
- Sum of squared deviations = 36 + 49 + 64 + 49 = 198
- SD = \(\sqrt{\frac{198}{4}} = \sqrt{49.5} \approx 7.03\)

3. **Dataset 3: 49, 17, 4, 18**
- Deviations: (49 - 22)^2 = 729, (17 - 22)^2 = 25, (4 - 22)^2 = 324, (18 - 22)^2 = 16
- Sum of squared deviations = 729 + 25 + 324 + 16 = 1094
- SD = \(\sqrt{\frac{1094}{4}} = \sqrt{273.5} \approx 16.53\)

4. **Dataset 4: 14, 20, 24, 30**
- Deviations: (14 - 19.5)^2 = 30.25, (20 - 19.5)^2 = 0.25, (24 - 19.5)^2 = 20.25, (30 - 19.5)^2 = 110.25
- Sum of squared deviations = 30.25 + 0.25 + 20.25 + 110.25 = 161.0
- SD = \(\sqrt{\frac{161}{4}} = \sqrt{40.25} \approx 6.35\)

### Summary of Standard Deviations:
- Dataset 1: ~10.86
- Dataset 2: ~7.03
- Dataset 3: ~16.53
- Dataset 4: ~6.35

### Conclusion:
The dataset that is most spread from its mean is **Dataset 3: 49, 17, 4, 18** with a standard deviation of approximately 16.53.