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QuestionFind the range, variance, and standard deviation for the data: 25, 33, 24, 40, 25, 39, 17, 45, 20, 38, 36, 37, 25, 44, 29, 36, 28, 40, 36, 33.

Studdy Solution

STEP 1

Assumptions1. The data set is a sample, not a population. The range has already been calculated as283. We are asked to find the variance and the standard deviation4. We are to use the shortcut formula for the unbiased estimator to compute the variance and standard deviation

STEP 2

First, we need to find the mean of the sample data. The mean is the sum of all values divided by the number of values.
Mean=Sum of all valuesNumber of valuesMean = \frac{Sum\, of\, all\, values}{Number\, of\, values}

STEP 3

Add up all the values in the data set.
Sum of all values=25+33+24+40+25+39+17+45+20+38+36+37+25+44+29+36+28+40+36+33Sum\, of\, all\, values =25 +33 +24 +40 +25 +39 +17 +45 +20 +38 +36 +37 +25 +44 +29 +36 +28 +40 +36 +33

STEP 4

Calculate the number of values in the data set.
Number of values=20Number\, of\, values =20

STEP 5

Calculate the mean of the data set.
Mean=Sum of all valuesNumber of valuesMean = \frac{Sum\, of\, all\, values}{Number\, of\, values}

STEP 6

Now that we have the mean, we can calculate the variance using the shortcut formula for the unbiased estimator. The formula isVariance=∑(xi−Mean)2n−1Variance = \frac{\sum (x_i - Mean)^2}{n -1}where xix_i are the data points, MeanMean is the mean of the data points, and nn is the number of data points.

STEP 7

Subtract the mean from each data point, square the result, and sum up all these squared values.
∑(xi−Mean)2=(25−Mean)2+(33−Mean)2+…+(36−Mean)2+(33−Mean)2\sum (x_i - Mean)^2 = (25 - Mean)^2 + (33 - Mean)^2 + \ldots + (36 - Mean)^2 + (33 - Mean)^2

STEP 8

Calculate the variance by dividing the sum of the squared deviations by the number of data points minus1.
Variance=∑(xi−Mean)2n−1Variance = \frac{\sum (x_i - Mean)^2}{n -1}

STEP 9

Now that we have the variance, we can find the standard deviation. The standard deviation is the square root of the variance.
Standard Deviation=VarianceStandard\, Deviation = \sqrt{Variance}

STEP 10

Calculate the standard deviation.
Standard Deviation=VarianceStandard\, Deviation = \sqrt{Variance}

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