QuestionFind the range, variance, and standard deviation of the data set: 23, 32, 31, 45, 25, 33, 44, 47, 37, 32, 37, 48, 47, 50, 22, 40, 30, 28, 22, 39. Use sample formulas.

Studdy Solution

STEP 1

Assumptions1. The data set provided is a sample, not a population. . The range is the difference between the highest and lowest values in the dataset.
3. The variance is a measure of how spread out the numbers in the data set are.
4. The standard deviation is the square root of the variance.
5. We will use the unbiased estimator formula for variance in a sample, which iss^ = \frac{\sum (x_i - \bar{x})^}{n-1}where $x_i$ are the data points, $\bar{x}$ is the mean of the data points, and $n$ is the number of data points.

STEP 2

First, we need to find the range. The range is the difference between the highest and lowest values in the dataset.
$Range = Max\, value - Min\, value$

STEP 3

Now, identify the maximum and minimum values in the dataset.
$Max\, value =50$ $Min\, value =22$

STEP 4

Substitute the maximum and minimum values into the formula to calculate the range.
$Range =50 -22$

STEP 5

Calculate the range.
$Range =50 -22 =28$

STEP 6

Next, we need to find the variance. First, calculate the mean of the data points.
$\bar{x} = \frac{\sum x_i}{n}$

STEP 7

Sum all the data points and divide by the number of data points to calculate the mean.
$\bar{x} = \frac{23 +32 +31 +45 +25 +33 +44 +47 +37 +32 +37 +48 +47 +50 +22 +40 +30 +28 +22 +39}{20}$

STEP 8

Calculate the mean.
$\bar{x} = \frac{23 +32 +31 +45 +25 +33 +44 +47 +37 +32 +37 +48 +47 +50 +22 +40 +30 +28 +22 +39}{20} =36.2$

STEP 9

Now, substitute the data points and the mean into the formula for variance.
$s^2 = \frac{\sum (x_i - \bar{x})^2}{n-}$

STEP 10

Calculate the variance.
$s^2 = \frac{(23-36.2)^2 + (32-36.2)^2 + \ldots + (22-36.2)^2 + (39-36.2)^2}{20-}$

STEP 11

Calculate the standard deviation. The standard deviation is the square root of the variance.
s = \sqrt{s^}

STEP 12

Substitute the variance into the formula to calculate the standard deviation.
$s = \sqrt{s^2}$

STEP 13

Calculate the standard deviation.
$s = \sqrt{s^2}$ The range is28, the variance is $s^2$ , and the standard deviation is $s$ .

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QuestionFind the range, variance, and standard deviation of the data set: 23, 32, 31, 45, 25, 33, 44, 47, 37, 32, 37, 48, 47, 50, 22, 40, 30, 28, 22, 39. Use sample formulas.

Studdy Solution

STEP 1

STEP 2

First, we need to find the range. The range is the difference between the highest and lowest values in the dataset.
$Range = Max\, value - Min\, value$

STEP 3

Now, identify the maximum and minimum values in the dataset.
$Max\, value =50$ $Min\, value =22$

STEP 4

Substitute the maximum and minimum values into the formula to calculate the range.
$Range =50 -22$

STEP 5

Calculate the range.
$Range =50 -22 =28$

STEP 6

Next, we need to find the variance. First, calculate the mean of the data points.
$\bar{x} = \frac{\sum x_i}{n}$

STEP 7

Sum all the data points and divide by the number of data points to calculate the mean.
$\bar{x} = \frac{23 +32 +31 +45 +25 +33 +44 +47 +37 +32 +37 +48 +47 +50 +22 +40 +30 +28 +22 +39}{20}$

STEP 8

Calculate the mean.
$\bar{x} = \frac{23 +32 +31 +45 +25 +33 +44 +47 +37 +32 +37 +48 +47 +50 +22 +40 +30 +28 +22 +39}{20} =36.2$

STEP 9

Now, substitute the data points and the mean into the formula for variance.
$s^2 = \frac{\sum (x_i - \bar{x})^2}{n-}$

STEP 10

Calculate the variance.
$s^2 = \frac{(23-36.2)^2 + (32-36.2)^2 + \ldots + (22-36.2)^2 + (39-36.2)^2}{20-}$

STEP 11

Calculate the standard deviation. The standard deviation is the square root of the variance.
s = \sqrt{s^}

STEP 12

Substitute the variance into the formula to calculate the standard deviation.
$s = \sqrt{s^2}$

STEP 13

Calculate the standard deviation.
$s = \sqrt{s^2}$ The range is28, the variance is $s^2$ , and the standard deviation is $s$ .

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QuestionFind the range, variance, and standard deviation of the data set: 23, 32, 31, 45, 25, 33, 44, 47, 37, 32, 37, 48, 47, 50, 22, 40, 30, 28, 22, 39. Use sample formulas.

Studdy Solution

STEP 1

STEP 2

First, we need to find the range. The range is the difference between the highest and lowest values in the dataset.Range=Max value−Min valueRange = Max\, value - Min\, valueRange=Maxvalue−Minvalue

STEP 3

Now, identify the maximum and minimum values in the dataset.Max value=50Max\, value =50Maxvalue=50Min value=22Min\, value =22Minvalue=22

STEP 4

Substitute the maximum and minimum values into the formula to calculate the range.Range=50−22Range =50 -22Range=50−22

STEP 5

Calculate the range.Range=50−22=28Range =50 -22 =28Range=50−22=28

STEP 6

Next, we need to find the variance. First, calculate the mean of the data points.xˉ=∑xin\bar{x} = \frac{\sum x_i}{n}xˉ=n∑xi​​

STEP 7

STEP 8

Calculate the mean.xˉ=23+32+31+45+25+33+44+47+37+32+37+48+47+50+22+40+30+28+22+3920=36.2\bar{x} = \frac{23 +32 +31 +45 +25 +33 +44 +47 +37 +32 +37 +48 +47 +50 +22 +40 +30 +28 +22 +39}{20} =36.2xˉ=2023+32+31+45+25+33+44+47+37+32+37+48+47+50+22+40+30+28+22+39​=36.2

STEP 9

Now, substitute the data points and the mean into the formula for variance.s2=∑(xi−xˉ)2n−s^2 = \frac{\sum (x_i - \bar{x})^2}{n-}s2=n−∑(xi​−xˉ)2​

STEP 10

Calculate the variance.s2=(23−36.2)2+(32−36.2)2+…+(22−36.2)2+(39−36.2)220−s^2 = \frac{(23-36.2)^2 + (32-36.2)^2 + \ldots + (22-36.2)^2 + (39-36.2)^2}{20-}s2=20−(23−36.2)2+(32−36.2)2+…+(22−36.2)2+(39−36.2)2​

STEP 11

Calculate the standard deviation. The standard deviation is the square root of the variance.s = \sqrt{s^}

STEP 12

Substitute the variance into the formula to calculate the standard deviation.s=s2s = \sqrt{s^2}s=s2​

STEP 13

Calculate the standard deviation.s=s2s = \sqrt{s^2}s=s2​The range is28, the variance is s2s^2s2, and the standard deviation is sss.

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First, we need to find the range. The range is the difference between the highest and lowest values in the dataset.
$Range = Max\, value - Min\, value$

Now, identify the maximum and minimum values in the dataset.
$Max\, value =50$ $Min\, value =22$

Substitute the maximum and minimum values into the formula to calculate the range.
$Range =50 -22$

Calculate the range.
$Range =50 -22 =28$

Next, we need to find the variance. First, calculate the mean of the data points.
$\bar{x} = \frac{\sum x_i}{n}$

Calculate the mean.
$\bar{x} = \frac{23 +32 +31 +45 +25 +33 +44 +47 +37 +32 +37 +48 +47 +50 +22 +40 +30 +28 +22 +39}{20} =36.2$

Now, substitute the data points and the mean into the formula for variance.
$s^2 = \frac{\sum (x_i - \bar{x})^2}{n-}$

Calculate the variance.
$s^2 = \frac{(23-36.2)^2 + (32-36.2)^2 + \ldots + (22-36.2)^2 + (39-36.2)^2}{20-}$

Calculate the standard deviation. The standard deviation is the square root of the variance.
s = \sqrt{s^}

Substitute the variance into the formula to calculate the standard deviation.
$s = \sqrt{s^2}$

Calculate the standard deviation.
$s = \sqrt{s^2}$ The range is28, the variance is $s^2$ , and the standard deviation is $s$ .