Math

QuestionFind the mean, median, mode, and standard deviation of these salaries: 98000, 115000, 75000, 88000, 65000, 107000, 72000, 71000, 88000, 57000, 88000, 73000, 97000, 85000, 87000, 73000, 93000, 88000, 71000, 65000, 87000, 83000, 75000, 44000, 81000, 76000, 81000, 89000.

Studdy Solution

STEP 1

Assumptions1. The list of salaries is given in the table. . We are asked to find the mean, median, mode, and standard deviation of the salaries.
3. All salaries are considered valid and there are no outliers.

STEP 2

First, we need to calculate the total sum of all salaries for finding the mean.Sum=i=1nSalaryiSum = \sum_{i=1}^{n} Salary_i

STEP 3

Now, add all the given salaries.
Sum=98000+115000+75000+88000+65000+107000+72000+71000+88000+57000+88000+73000+97000+85000+87000+73000+93000+88000+71000+65000+87000+83000+75000+44000+81000+76000+81000+89000Sum =98000 +115000 +75000 +88000 +65000 +107000 +72000 +71000 +88000 +57000 +88000 +73000 +97000 +85000 +87000 +73000 +93000 +88000 +71000 +65000 +87000 +83000 +75000 +44000 +81000 +76000 +81000 +89000

STEP 4

Calculate the sum of all salaries.
Sum=2107000Sum =2107000

STEP 5

Now, calculate the mean of the salaries by dividing the sum by the total number of salaries.
Mean=SumnMean = \frac{Sum}{n}

STEP 6

Plug in the values for the sum and the total number of salaries to calculate the mean.
Mean=210700028Mean = \frac{2107000}{28}

STEP 7

Calculate the mean salary.
Mean=75250Mean =75250

STEP 8

Next, we need to find the median of the salaries. To do this, we first need to sort the salaries in ascending order.

STEP 9

After sorting the salaries, if the number of salaries is even, the median is the average of the two middle numbers. If the number of salaries is odd, the median is the middle number.

STEP 10

Since the number of salaries is28 (an even number), we find the average of the14th and15th salaries.
Median=14thsalary+15thsalary2Median = \frac{14th\, salary +15th\, salary}{2}

STEP 11

Plug in the values for the14th and15th salaries to calculate the median.
Median=81000+83000Median = \frac{81000 +83000}{}

STEP 12

Calculate the median salary.
Median=82000Median =82000

STEP 13

Next, we need to find the mode of the salaries. The mode is the salary that appears most frequently.

STEP 14

From the list of salaries, we see that $88000 appears most frequently, so it is the mode.

STEP 15

Finally, we need to calculate the standard deviation of the salaries. The standard deviation is a measure of how spread out the numbers are from the mean.
StandardDeviation=i=n(SalaryiMean)2nStandard\, Deviation = \sqrt{\frac{\sum_{i=}^{n} (Salary_i - Mean)^2}{n}}

STEP 16

First, calculate the sum of the squared differences from the mean.
i=n(SalaryiMean)2=(9800075250)2+(11500075250)2++(8900075250)2\sum_{i=}^{n} (Salary_i - Mean)^2 = (98000 -75250)^2 + (115000 -75250)^2 + \ldots + (89000 -75250)^2

STEP 17

Calculate the sum of the squared differences from the mean.
i=n(SalaryiMean)2=1616250000\sum_{i=}^{n} (Salary_i - Mean)^2 =1616250000

STEP 18

Now, calculate the standard deviation by plugging in the values for the sum of the squared differences from the mean and the total number of salaries.
StandardDeviation=161625000028Standard\, Deviation = \sqrt{\frac{1616250000}{28}}

STEP 19

Calculate the standard deviation.
StandardDeviation=21357Standard\, Deviation =21357So, the mean of the salaries is 75250,themedianis75250, the median is 82000, the mode is 88000,andthestandarddeviationis88000, and the standard deviation is 21357.

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