Solution - Statistics

Divide the sum by the number of terms:

Sum
$$\frac{27753}{40}$$
Number of terms
$$4$$

$$x̄=\frac{27753}{160}=173.456$$

The mean equals $$173.456$$

Arrange the numbers in ascending order:
150,165,182.325,196.5

Count the number of terms:
There are (4) terms

Because there is an even number of terms, identify the middle two terms:
150,165,182.325,196.5

Find the value that is halfway between the middle two terms by adding them together and dividing by 2:
$$\left(165+182.325\right)/2=347.325/2=173.6625$$

The median equals 173.6625

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

The highest value equals 196.5
The lowest value equals 150

$$196.5-150=46.5$$

The range equals 46.5

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 173.456

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:
$$550.196+71.508+78.655+531.014=1231.373$$
Number of terms:
$$4$$
Number of terms minus 1:
3

Variance:
$$\frac{1231.373}{3}=410.458$$

The sample variance ($$s^2$$) equals 410.458

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: $$s^2=410.458$$

Find the square root:
$$s=\sqrt{\left(410.458\right)}=20.260$$

The standard deviation ($$s$$) equals 20.26