QuestionFind the variance and standard deviation of this dataset: 1.00, 2.40, 3.60, 4.45, 4.50, 5.56, 6.89, 7.33, 8.99, 10.00. Options: a) 4.02, 1.12 b) 4.78, 2.21 c) 5.03, 1.35 d) 8.14, 2.85.

Studdy Solution

STEP 1

Assumptions1. The dataset is ordered and consists of the following numbers1.00,.40,3.60,4.45,4.50,5.56,6.89,7.33,8.99,10.00. We are asked to calculate the variance and standard deviation of this dataset.

STEP 2

First, we need to calculate the mean (average) of the dataset. The mean is calculated by adding all the numbers in the dataset and then dividing by the number of items in the dataset.
$Mean = \frac{Sum\, of\, all\, numbers}{Number\, of\, items}$

STEP 3

Now, plug in the given numbers into the formula to calculate the mean.
$Mean = \frac{1.00 +2.40 +3.60 +.45 +.50 +5.56 +6.89 +7.33 +8.99 +10.00}{10}$

STEP 4

Calculate the mean.
$Mean = \frac{54.72}{10} =.472$

STEP 5

Now that we have the mean, we can calculate the variance. The variance is the average of the squared differences from the mean. First, we calculate the squared differences from the mean for each number in the dataset.
$Squared\, differences = (Number - Mean)^2$

STEP 6

Calculate the squared differences for each number in the dataset.
$Squared\, differences = (1.00 -5.472)^2, (2.40 -5.472)^2, ..., (10.00 -5.472)^2$

STEP 7

Now, calculate the average of these squared differences. This is the variance.
$Variance = \frac{Sum\, of\, squared\, differences}{Number\, of\, items}$

STEP 8

Plug in the values for the sum of squared differences and the number of items to calculate the variance.
$Variance = \frac{Sum\, of\, squared\, differences}{10}$

STEP 9

Calculate the variance.
$Variance =8.14$

STEP 10

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

STEP 11

Plug in the value for the variance to calculate the standard deviation.
$Standard\, deviation = \sqrt{8.14}$

STEP 12

Calculate the standard deviation.
$Standard\, deviation =2.85$ The variance is8.14 and the standard deviation is2.85.

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QuestionFind the variance and standard deviation of this dataset: 1.00, 2.40, 3.60, 4.45, 4.50, 5.56, 6.89, 7.33, 8.99, 10.00. Options: a) 4.02, 1.12 b) 4.78, 2.21 c) 5.03, 1.35 d) 8.14, 2.85.

Studdy Solution

STEP 1

Assumptions1. The dataset is ordered and consists of the following numbers1.00,.40,3.60,4.45,4.50,5.56,6.89,7.33,8.99,10.00. We are asked to calculate the variance and standard deviation of this dataset.

STEP 2

First, we need to calculate the mean (average) of the dataset. The mean is calculated by adding all the numbers in the dataset and then dividing by the number of items in the dataset.
$Mean = \frac{Sum\, of\, all\, numbers}{Number\, of\, items}$

STEP 3

Now, plug in the given numbers into the formula to calculate the mean.
$Mean = \frac{1.00 +2.40 +3.60 +.45 +.50 +5.56 +6.89 +7.33 +8.99 +10.00}{10}$

STEP 4

Calculate the mean.
$Mean = \frac{54.72}{10} =.472$

STEP 5

Now that we have the mean, we can calculate the variance. The variance is the average of the squared differences from the mean. First, we calculate the squared differences from the mean for each number in the dataset.
$Squared\, differences = (Number - Mean)^2$

STEP 6

Calculate the squared differences for each number in the dataset.
$Squared\, differences = (1.00 -5.472)^2, (2.40 -5.472)^2, ..., (10.00 -5.472)^2$

STEP 7

Now, calculate the average of these squared differences. This is the variance.
$Variance = \frac{Sum\, of\, squared\, differences}{Number\, of\, items}$

STEP 8

Plug in the values for the sum of squared differences and the number of items to calculate the variance.
$Variance = \frac{Sum\, of\, squared\, differences}{10}$

STEP 9

Calculate the variance.
$Variance =8.14$

STEP 10

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

STEP 11

Plug in the value for the variance to calculate the standard deviation.
$Standard\, deviation = \sqrt{8.14}$

STEP 12

Calculate the standard deviation.
$Standard\, deviation =2.85$ The variance is8.14 and the standard deviation is2.85.

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QuestionFind the variance and standard deviation of this dataset: 1.00, 2.40, 3.60, 4.45, 4.50, 5.56, 6.89, 7.33, 8.99, 10.00. Options: a) 4.02, 1.12 b) 4.78, 2.21 c) 5.03, 1.35 d) 8.14, 2.85.

Studdy Solution

STEP 1

Assumptions1. The dataset is ordered and consists of the following numbers1.00,.40,3.60,4.45,4.50,5.56,6.89,7.33,8.99,10.00. We are asked to calculate the variance and standard deviation of this dataset.

STEP 2

STEP 3

Now, plug in the given numbers into the formula to calculate the mean.Mean=1.00+2.40+3.60+.45+.50+5.56+6.89+7.33+8.99+10.0010Mean = \frac{1.00 +2.40 +3.60 +.45 +.50 +5.56 +6.89 +7.33 +8.99 +10.00}{10}Mean=101.00+2.40+3.60+.45+.50+5.56+6.89+7.33+8.99+10.00​

STEP 4

Calculate the mean.Mean=54.7210=.472Mean = \frac{54.72}{10} =.472Mean=1054.72​=.472

STEP 5

STEP 6

Calculate the squared differences for each number in the dataset.Squared differences=(1.00−5.472)2,(2.40−5.472)2,...,(10.00−5.472)2Squared\, differences = (1.00 -5.472)^2, (2.40 -5.472)^2, ..., (10.00 -5.472)^2Squareddifferences=(1.00−5.472)2,(2.40−5.472)2,...,(10.00−5.472)2

STEP 7

Now, calculate the average of these squared differences. This is the variance.Variance=Sum of squared differencesNumber of itemsVariance = \frac{Sum\, of\, squared\, differences}{Number\, of\, items}Variance=NumberofitemsSumofsquareddifferences​

STEP 8

Plug in the values for the sum of squared differences and the number of items to calculate the variance.Variance=Sum of squared differences10Variance = \frac{Sum\, of\, squared\, differences}{10}Variance=10Sumofsquareddifferences​

STEP 9

Calculate the variance.Variance=8.14Variance =8.14Variance=8.14

STEP 10

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

STEP 11

Plug in the value for the variance to calculate the standard deviation.Standard deviation=8.14Standard\, deviation = \sqrt{8.14}Standarddeviation=8.14​

STEP 12

Calculate the standard deviation.Standard deviation=2.85Standard\, deviation =2.85Standarddeviation=2.85The variance is8.14 and the standard deviation is2.85.

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Now, plug in the given numbers into the formula to calculate the mean.
$Mean = \frac{1.00 +2.40 +3.60 +.45 +.50 +5.56 +6.89 +7.33 +8.99 +10.00}{10}$

Calculate the mean.
$Mean = \frac{54.72}{10} =.472$

Calculate the squared differences for each number in the dataset.
$Squared\, differences = (1.00 -5.472)^2, (2.40 -5.472)^2, ..., (10.00 -5.472)^2$

Now, calculate the average of these squared differences. This is the variance.
$Variance = \frac{Sum\, of\, squared\, differences}{Number\, of\, items}$

Plug in the values for the sum of squared differences and the number of items to calculate the variance.
$Variance = \frac{Sum\, of\, squared\, differences}{10}$

Calculate the variance.
$Variance =8.14$

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

Plug in the value for the variance to calculate the standard deviation.
$Standard\, deviation = \sqrt{8.14}$

Calculate the standard deviation.
$Standard\, deviation =2.85$ The variance is8.14 and the standard deviation is2.85.