Question

If the outlier is excluded from the following data set, what happens to the mean?
110, 112, 109, 112, 21, 108, 110, 111, 108

It does not change.

It depends whether the outlier is a high outlier or a low outlier.

It increases.

It decreases.

Answer 1

To analyze what happens to the mean when the outlier is excluded from the data set, let's first identify the outlier in the provided data:

Given data set: 110, 112, 109, 112, 21, 108, 110, 111, 108

Here, the number **21** is significantly lower than the rest of the numbers, which are around 108 to 112. Therefore, **21** is the outlier.

Now, let's calculate the mean of the original data set:

1. Calculate the sum of the data:
\( 110 + 112 + 109 + 112 + 21 + 108 + 110 + 111 + 108 = 821 \)

2. Count the number of values (n):
There are 9 numbers in the dataset.

3. Calculate the mean:
\( \text{Mean} = \frac{821}{9} \approx 91.22 \)

Now, let's exclude the outlier (21) and recalculate the mean.

1. Calculate the new sum without the outlier:
\( 110 + 112 + 109 + 112 + 108 + 110 + 111 + 108 = 870 \)

2. Count the number of values now (n):
There are 8 numbers remaining.

3. Calculate the new mean:
\( \text{New Mean} = \frac{870}{8} = 108.75 \)

Now comparing the two means:
- Original Mean: Approximately 91.22
- New Mean: 108.75

Excluding the outlier increases the mean.
Thus, the answer is: **It increases.**