## Mean, Median,Mode

The total blood cholesterol level was measured for each of 21adults. Here are the 21 measurements (in mg/dL). 129, 144, 159, 162, 163, 168, 183, 184, 184, 185, 186, 195, 196, 197, 198, 207, 209, 221, 222, 234, 246.(a) Which measures of central tendency do not exist for this data set? Choose all that apply.

Mean

Median

Mode

None of these measures

(b) Suppose that the measurement 129 (the smallest measurement in the data set) were replaced by 52. Which measures of central tendency would be affected by the change? Choose all that apply.

Mean

Median

Mode

None of these measures

(c) Suppose that, starting with the original data set, the smallest measurement were removed. Which measures of central tendency would be changed from those of the original data set? Choose all that apply.

Mean

Median

Mode

None of these measures

(d) Which of the following best describes the distribution of the original data? Choose only one.

Negatively skewed

Positively skewed

Roughly symmetrical

(a) The mean and median do not exist for this data set. The mean is not defined because there is an even number of data points and the median is not defined because the data points are not arranged in increasing order. The mode is 184, which is the most frequent data point.

(b) The mean and median would be affected by the change. The mean would decrease because 52 is less than 129. The median would also decrease because 52 would be the new smallest data point. The mode would not be affected because 52 is not the most frequent data point.

(c) The mean and median would not be changed if the smallest measurement were removed. The mean is not affected by removing a single data point. The median would not be affected because the data points would still be arranged in increasing order after the smallest measurement is removed. The mode might be affected because the most frequent data point might change.

(d) The distribution of the original data is negatively skewed. This is because the tail of the distribution is longer on the left side, which means that there are more data points with lower values than there are data points with higher values.

Here is a more detailed explanation of each answer:

(a) The mean is not defined because there is an even number of data points (21). When there is an even number of data points, the mean is calculated by taking the sum of the data points and dividing by 2. However, in this case, the sum of the data points is not even divisible by 2. Therefore, the mean is not defined.

The median is not defined because the data points are not arranged in increasing order. The median is the middle data point when the data points are arranged in increasing order. However, in this case, the data points are not arranged in increasing order, so there is no middle data point.

The mode is 184, which is the most frequent data point. There are 4 data points with a value of 184.

(b) The mean would decrease because 52 is less than 129. The mean is calculated by taking the sum of the data points and dividing by the number of data points. If we replace 129 with 52, the sum of the data points will decrease, which will cause the mean to decrease.

The median would also decrease because 52 would be the new smallest data point. The median is the middle data point when the data points are arranged in increasing order. If we replace 129 with 52, the data points will still be arranged in increasing order, but the new middle data point will be 52.

The mode would not be affected because 52 is not the most frequent data point. The mode is the most frequent data point. If we replace 129 with 52, the most frequent data point will still be 184.

(c) The mean and median would not be changed if the smallest measurement were removed. The mean is not affected by removing a single data point. The median would not be affected because the data points would still be arranged in increasing order after the smallest measurement is removed.

The mode might be affected because the most frequent data point might change. If the smallest measurement is 129, then the most frequent data point is 184. However, if the smallest measurement is removed, then the most frequent data point might change to another value.

(d) The distribution of the original data is negatively skewed. This is because the tail of the distribution is longer on the left side, which means that there are more data points with lower values than there are data points with higher values.

The distribution of the data can be visualized by creating a histogram. A histogram is a graph that shows the distribution of data by dividing the data into bins and then plotting the number of data points in each bin.

The following histogram shows the distribution of the original data:

The histogram shows that there are more data points with values below 180 than there are data points with values above 180. This means that the tail of the distribution is longer on the left side, which indicates that the distribution is negatively skewed.