QuestionCalculate the standard deviation and variance of the data: 16.3, 1.7, 28, 9.2, 15.2, 26.1, 8.5, 20.4.

Studdy Solution

STEP 1

Assumptions1. The data provided is a sample from a larger population. . We are asked to calculate the sample standard deviation and variance, not the population standard deviation and variance.

STEP 2

First, we need to calculate the mean (average) of the sample data. The mean is calculated by summing all the data points and dividing by the number of data points.
$\bar{x} = \frac{\sum x}{n}$

STEP 3

Now, plug in the given values for the data points to calculate the mean.
$\bar{x} = \frac{16.3 +1.7 +28 +9.2 +15.2 +26.1 +8.5 +20.}{8}$

STEP 4

Calculate the mean.
$\bar{x} = \frac{125.4}{8} =15.675$

STEP 5

Next, we need to calculate the variance. The variance is the average of the squared differences from the mean. It is calculated using the following formula $s^2 = \frac{\sum (x - \bar{x})^2}{n -1}$

STEP 6

Now, plug in the given values for the data points and the mean to calculate the variance.
$s^2 = \frac{(16.3 -15.675)^2 + (1. -15.675)^2 + (28 -15.675)^2 + (9.2 -15.675)^2 + (15.2 -15.675)^2 + (26.1 -15.675)^2 + (8.5 -15.675)^2 + (20.4 -15.675)^2}{8 -1}$

STEP 7

Calculate the variance.
$s^2 = \frac{0.39 +196.56 +151.78 +41.86 +0.23 +108.18 +51.31 +22.41}{7} = \frac{572.72}{7} =81.82$

STEP 8

Finally, we need to calculate the standard deviation. The standard deviation is the square root of the variance.
$s = \sqrt{s^2}$

STEP 9

Now, plug in the value for the variance to calculate the standard deviation.
$s = \sqrt{81.82}$

STEP 10

Calculate the standard deviation.
$s =9.05$ The standard deviation of the sample data is9.05 and the variance is81.82.

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