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

Studdy Solution

STEP 1

Assumptions1. The data represents samples, not a population. . We are asked to find the range, variance, and standard deviation.
3. The given data is $\begin{array}{llllllllll} 23 &32 &31 &45 &25 &33 &44 &47 &37 &32 \\ 37 &48 &47 &50 &22 &40 &30 &28 &22 &39\end{array}$

STEP 2

First, let's find the range of the data. The range is the difference between the highest and lowest values in the dataset.
$Range = Maximum\, value - Minimum\, value$

STEP 3

Identify the maximum and minimum values from the given dataset.
$Maximum\, value =50$ $Minimum\, value =22$

STEP 4

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

STEP 5

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

STEP 6

Now, let's calculate the variance. The variance is a measure of how much the values in the dataset vary from the mean.The formula for the variance of a sample is $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$ Where- $x_i$ are the sample observations- $\bar{x}$ is the sample mean- $n$ is the number of observations

STEP 7

First, calculate the mean of the dataset. The mean is the sum of all the values divided by the number of values.
$\bar{x} = \frac{\sum x_i}{n}$

STEP 8

Calculate the sum of the dataset and the number of observations.
$\sum x_i =23 +32 +31 +45 +25 +33 +44 +47 +37 +32 +37 +48 +47 +50 +22 +40 +30 +28 +22 +39 =732$ $n =20$

STEP 9

Substitute the sum and the number of observations into the mean formula.
$\bar{x} = \frac{732}{20}$

STEP 10

Calculate the mean.
$\bar{x} = \frac{732}{20} =36.6$

STEP 11

Now, substitute the mean, the observations, and the number of observations into the variance formula.
s^ = \frac{(23-36.6)^ + (32-36.6)^ + \ldots + (22-36.6)^ + (39-36.6)^}{20-}

STEP 12

Calculate the variance.

STEP 13

The standard deviation is the square root of the variance.
$s = \sqrt{s^2}$

STEP 14

Substitute the variance into the standard deviation formula.

STEP 15

Calculate the standard deviation.

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

Studdy Solution

STEP 1

Assumptions1. The data represents samples, not a population. . We are asked to find the range, variance, and standard deviation.
3. The given data is $\begin{array}{llllllllll} 23 &32 &31 &45 &25 &33 &44 &47 &37 &32 \\ 37 &48 &47 &50 &22 &40 &30 &28 &22 &39\end{array}$

STEP 2

First, let's find the range of the data. The range is the difference between the highest and lowest values in the dataset.
$Range = Maximum\, value - Minimum\, value$

STEP 3

Identify the maximum and minimum values from the given dataset.
$Maximum\, value =50$ $Minimum\, value =22$

STEP 4

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

STEP 5

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

STEP 6

STEP 7

First, calculate the mean of the dataset. The mean is the sum of all the values divided by the number of values.
$\bar{x} = \frac{\sum x_i}{n}$

STEP 8

Calculate the sum of the dataset and the number of observations.
$\sum x_i =23 +32 +31 +45 +25 +33 +44 +47 +37 +32 +37 +48 +47 +50 +22 +40 +30 +28 +22 +39 =732$ $n =20$

STEP 9

Substitute the sum and the number of observations into the mean formula.
$\bar{x} = \frac{732}{20}$

STEP 10

Calculate the mean.
$\bar{x} = \frac{732}{20} =36.6$

STEP 11

Now, substitute the mean, the observations, and the number of observations into the variance formula.
s^ = \frac{(23-36.6)^ + (32-36.6)^ + \ldots + (22-36.6)^ + (39-36.6)^}{20-}

STEP 12

Calculate the variance.

STEP 13

The standard deviation is the square root of the variance.
$s = \sqrt{s^2}$

STEP 14

Substitute the variance into the standard deviation formula.

STEP 15

Calculate the standard deviation.

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

Studdy Solution

STEP 1

STEP 2

First, let's find the range of the data. The range is the difference between the highest and lowest values in the dataset.Range=Maximum value−Minimum valueRange = Maximum\, value - Minimum\, valueRange=Maximumvalue−Minimumvalue

STEP 3

Identify the maximum and minimum values from the given dataset.Maximum value=50Maximum\, value =50Maximumvalue=50Minimum value=22Minimum\, value =22Minimumvalue=22

STEP 4

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

STEP 5

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

STEP 6

STEP 7

First, calculate the mean of the dataset. The mean is the sum of all the values divided by the number of values.xˉ=∑xin\bar{x} = \frac{\sum x_i}{n}xˉ=n∑xi​​

STEP 8

STEP 9

Substitute the sum and the number of observations into the mean formula.xˉ=73220\bar{x} = \frac{732}{20}xˉ=20732​

STEP 10

Calculate the mean.xˉ=73220=36.6\bar{x} = \frac{732}{20} =36.6xˉ=20732​=36.6

STEP 11

Now, substitute the mean, the observations, and the number of observations into the variance formula.s^ = \frac{(23-36.6)^ + (32-36.6)^ + \ldots + (22-36.6)^ + (39-36.6)^}{20-}

STEP 12

Calculate the variance.

STEP 13

The standard deviation is the square root of the variance.s=s2s = \sqrt{s^2}s=s2​

STEP 14

Substitute the variance into the standard deviation formula.

STEP 15

Calculate the standard deviation.

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First, let's find the range of the data. The range is the difference between the highest and lowest values in the dataset.
$Range = Maximum\, value - Minimum\, value$

Identify the maximum and minimum values from the given dataset.
$Maximum\, value =50$ $Minimum\, value =22$

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

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

First, calculate the mean of the dataset. The mean is the sum of all the values divided by the number of values.
$\bar{x} = \frac{\sum x_i}{n}$

Substitute the sum and the number of observations into the mean formula.
$\bar{x} = \frac{732}{20}$

Calculate the mean.
$\bar{x} = \frac{732}{20} =36.6$

Now, substitute the mean, the observations, and the number of observations into the variance formula.
s^ = \frac{(23-36.6)^ + (32-36.6)^ + \ldots + (22-36.6)^ + (39-36.6)^}{20-}

The standard deviation is the square root of the variance.
$s = \sqrt{s^2}$