Question

Select the outlier in the data set.
If the outlier were removed from the data set, would the mean increase or decrease?


53 59 686 62 52 38 47 81 68 50 57

Answer 1

To identify the outlier in the data set \( \{53, 59, 686, 62, 52, 38, 47, 81, 68, 50, 57\} \), we can observe that the number 686 is significantly larger than the other values, which range from 38 to 81. Thus, 686 is the outlier.

Now let's determine how removing the outlier affects the mean.

1. **Calculate the mean of the original data set**:
\[
\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}
\]

- The sum of the values:
\[
53 + 59 + 686 + 62 + 52 + 38 + 47 + 81 + 68 + 50 + 57 = 1,052
\]
- The number of values:
\[
11
\]
- Therefore, the mean is:
\[
\text{Mean} = \frac{1,052}{11} \approx 95.64
\]

2. **Remove the outlier and calculate the new mean**:
The new data set (without the outlier 686) is \( \{53, 59, 62, 52, 38, 47, 81, 68, 50, 57\} \).

- The new sum of the values:
\[
53 + 59 + 62 + 52 + 38 + 47 + 81 + 68 + 50 + 57 = 417
\]
- The number of values now:
\[
10
\]
- Therefore, the new mean is:
\[
\text{New Mean} = \frac{417}{10} = 41.7
\]

3. **Comparison of the means**:
- Original mean: \( \approx 95.64 \)
- New mean (without outlier): \( 41.7 \)

Removing the outlier (686) causes the mean to **decrease**.

Thus, the outlier is 686, and removing it would cause the mean to decrease.