Solution - Statistics

Divide the sum by the number of terms:

Sum
$$517$$
Number of terms
$$5$$

$$x̄=\frac{517}{5}=103.4$$

The mean equals $$103.4$$

Arrange the numbers in ascending order:
22,54,108,135,198

Count the number of terms:
There are (5) terms

Because there is an odd number of terms, the middle term is the median:
22,54,108,135,198

The median equals 108

To find the range, subtract the lowest value from the highest value.

The highest value equals 198
The lowest value equals 22

$$198-22=176$$

The range equals 176

To find the sample variance, find the difference between each term and the mean, square the results, add together all of the squared results, and divide the sum by the number of terms minus 1.

The mean equals 103.4

To get the squared differences, subtract the mean from each term and square the result:

To get the sample variance, add together the squared differences and divide their sum by the number of terms minus 1

Sum:
$$8949.16+998.56+21.16+2440.36+6625.96=19035.20$$
Number of terms:
$$5$$
Number of terms minus 1:
4

Variance:
$$\frac{19035.20}{4}=4758.8$$

The sample variance ($$s^2$$) equals 4758.8

The standard deviation of the sample equals the square root of the sample variance. This is why the variance is usually represented by a squared variable.

Variance: $$s^2=4758.8$$

Find the square root:
$$s=\sqrt{\left(4758.8\right)}=68.984$$

The standard deviation ($$s$$) equals 68.984