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QuestionCalculate the variance for the numbers 20.7,33.2,21.5,58,23.8,110,30.6,24,74,60.8,40.7,45.5,65.620.7, 33.2, 21.5, 58, 23.8, 110, 30.6, 24, 74, 60.8, 40.7, 45.5, 65.6. Indicate if it's sample or population variance.

Studdy Solution

STEP 1

Assumptions1. The given set of numbers is 20.7,33.,21.5,58,23.8,110,30.6,24,74,60.8,40.7,45.5,65.620.7,33.,21.5,58,23.8,110,30.6,24,74,60.8,40.7,45.5,65.6. . We are asked to find the variance of this set of numbers.
3. We are not told whether to find the sample variance or the population variance, so we will find both.

STEP 2

First, we need to find the mean (average) of the given set of numbers. The mean is calculated by summing all the numbers in the set and then dividing by the count of numbers in the set.
Mean=Sum of all numbersCount of numbersMean = \frac{Sum\, of\, all\, numbers}{Count\, of\, numbers}

STEP 3

Now, plug in the given values to calculate the mean.
Mean=20.7+33.2+21.5+58+23.8+110+30.6+24+74+60.8+40.7+45.5+65.613Mean = \frac{20.7 +33.2 +21.5 +58 +23.8 +110 +30.6 +24 +74 +60.8 +40.7 +45.5 +65.6}{13}

STEP 4

Calculate the mean.
Mean=608.413=46.8Mean = \frac{608.4}{13} =46.8

STEP 5

Next, we need to find the squared differences from the mean for each number in the set. This is calculated by subtracting the mean from each number, squaring the result, and then summing all these squared differences.
Sum of squared differences=∑i=1n(xi−Mean)2Sum\, of\, squared\, differences = \sum_{i=1}^{n} (x_i - Mean)^2

STEP 6

Now, plug in the given values and the calculated mean to find the sum of squared differences.
Sum of squared differences=(20.−46.8)2+(33.2−46.8)2+...+(65.6−46.8)2Sum\, of\, squared\, differences = (20. -46.8)^2 + (33.2 -46.8)^2 + ... + (65.6 -46.8)^2

STEP 7

Calculate the sum of squared differences.
Sum of squared differences=681.64+184.96+...+353.44=4872.04Sum\, of\, squared\, differences =681.64 +184.96 + ... +353.44 =4872.04

STEP 8

Now, we can find the population variance. The population variance is calculated by dividing the sum of squared differences by the count of numbers in the set.
Population variance=Sum of squared differencesCount of numbersPopulation\, variance = \frac{Sum\, of\, squared\, differences}{Count\, of\, numbers}

STEP 9

Plug in the values for the sum of squared differences and the count of numbers to calculate the population variance.
Population variance=4872.0413Population\, variance = \frac{4872.04}{13}

STEP 10

Calculate the population variance.
Population variance=4872.0413=374.77Population\, variance = \frac{4872.04}{13} =374.77

STEP 11

Now, we can find the sample variance. The sample variance is calculated by dividing the sum of squared differences by the count of numbers in the set minus.
Sample variance=Sum of squared differencesCount of numbers−Sample\, variance = \frac{Sum\, of\, squared\, differences}{Count\, of\, numbers -}

STEP 12

Plug in the values for the sum of squared differences and the count of numbers to calculate the sample variance.
Sample variance=4872.04−Sample\, variance = \frac{4872.04}{ -}

STEP 13

Calculate the sample variance.
Sample variance=4872.0412=406.00Sample\, variance = \frac{4872.04}{12} =406.00The population variance of the given set of numbers is374.77 and the sample variance is406.00.

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