Answer:

To find the mean, median, mode, variance, and standard deviation of the given data set:

Data set: 15, 19, 22, 21, 23, 19, 17, 16, 22, 17, 19, 21, 23, 21, 20, 23, 17, 18, 18, 22

1. Mean:

To find the mean, add up all the numbers in the data set and divide the sum by the total number of values.

Mean = (15 + 19 + 22 + 21 + 23 + 19 + 17 + 16 + 22 + 17 + 19 + 21 + 23 + 21 + 20 + 23 + 17 + 18 + 18 + 22) / 20

Mean = 20.9

2. Median:

To find the median, arrange the numbers in ascending order and find the middle value. If there are two middle values, find their average.

Arranged data set: 15, 16, 17, 17, 17, 18, 18, 19, 19, 19, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23

Median = (19 + 20) / 2

Median = 19.5

3. Mode:

The mode is the value(s) that appear most frequently in the data set.

Mode = 17, 19, 21, 22, 23 (all appear twice)

4. Variance:

Variance measures the spread or dispersion of the data set. It is calculated by finding the average of the squared differences between each value and the mean.

Variance = [(15 - 20.9)^2 + (19 - 20.9)^2 + ... + (18 - 20.9)^2 + (22 - 20.9)^2] / 20

Variance = 7.89

5. Standard Deviation:

The standard deviation is the square root of the variance and provides a measure of how spread out the data is.

Standard Deviation = √Variance

Standard Deviation = √7.89

Standard Deviation ≈ 2.81

To summarize:

1. Mean = 20.9

2. Median = 19.5

3. Mode = 17, 19, 21, 22, 23

4. Variance = 7.89

5. Standard Deviation ≈ 2.81