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QuestionDetermine which statistic (mean, median, or mode) changes most if one of the 80's in {80,39,35,40,80,50}\{80, 39, 35, 40, 80, 50\} is replaced with 50.

Studdy Solution

STEP 1

Assumptions1. The given set of numbers is {80,39,35,40,80,50}\{80,39,35,40,80,50\} . We are replacing one of the80's with a503. We are comparing the changes in mean, median, and mode after the replacement

STEP 2

First, we need to calculate the mean, median, and mode of the original set of numbers.
The mean is calculated as the sum of all numbers divided by the count of numbers.
Mean=Sum of all numbersCount of numbersMean = \frac{Sum\, of\, all\, numbers}{Count\, of\, numbers}

STEP 3

Now, plug in the given values to calculate the mean.
Mean=80+39+35+40+80+506Mean = \frac{80 +39 +35 +40 +80 +50}{6}

STEP 4

Calculate the mean.
Mean=3246=54Mean = \frac{324}{6} =54

STEP 5

The median is the middle number when the numbers are arranged in ascending order. If there is an even number of numbers, the median is the average of the two middle numbers.
The given set of numbers in ascending order is {35,39,40,50,80,80}\{35,39,40,50,80,80\}. So, the median is the average of the third and fourth numbers.
Median=40+502Median = \frac{40 +50}{2}

STEP 6

Calculate the median.
Median=40+502=45Median = \frac{40 +50}{2} =45

STEP 7

The mode is the number that appears most frequently in the set. In this case, the mode is80 as it appears twice.

STEP 8

Next, we need to calculate the mean, median, and mode after replacing one of the80's with a50.
The new set of numbers is {50,39,35,40,80,50}\{50,39,35,40,80,50\}.
The new mean is calculated as the sum of all numbers divided by the count of numbers.
New Mean=Sum of all numbersCount of numbersNew\, Mean = \frac{Sum\, of\, all\, numbers}{Count\, of\, numbers}

STEP 9

Now, plug in the given values to calculate the new mean.
New Mean=50+39+35+40+80+506New\, Mean = \frac{50 +39 +35 +40 +80 +50}{6}

STEP 10

Calculate the new mean.
New Mean=2946=49New\, Mean = \frac{294}{6} =49

STEP 11

The new set of numbers in ascending order is {35,39,40,50,50,80}\{35,39,40,50,50,80\}. So, the new median is the average of the third and fourth numbers.
New Median=40+50New\, Median = \frac{40 +50}{}

STEP 12

Calculate the new median.
New Median=40+502=45New\, Median = \frac{40 +50}{2} =45

STEP 13

The new mode is the number that appears most frequently in the set. In this case, the mode is50 as it appears twice.

STEP 14

Now, we can calculate the changes in mean, median, and mode after the replacement.
The change in mean is the absolute difference between the original mean and the new mean.
Change in Mean=∣Mean−New Mean∣Change\, in\, Mean = |Mean - New\, Mean|

STEP 15

Plug in the values for the mean and the new mean to calculate the change in mean.
Change in Mean=∣54−49∣Change\, in\, Mean = |54 -49|

STEP 16

Calculate the change in mean.
Change in Mean=∣54−49∣=5Change\, in\, Mean = |54 -49| =5

STEP 17

The change in median is the absolute difference between the original median and the new median.
Change in Median=∣Median−New Median∣Change\, in\, Median = |Median - New\, Median|

STEP 18

Plug in the values for the median and the new median to calculate the change in median.
Change in Median=∣45−45∣Change\, in\, Median = |45 -45|

STEP 19

Calculate the change in median.
Change in Median=∣45−45∣=Change\, in\, Median = |45 -45| =

STEP 20

The change in mode is the absolute difference between the original mode and the new mode.
Change in Mode=∣Mode−New Mode∣Change\, in\, Mode = |Mode - New\, Mode|

STEP 21

Plug in the values for the mode and the new mode to calculate the change in mode.
Change in Mode=∣80−50∣Change\, in\, Mode = |80 -50|

STEP 22

Calculate the change in mode.
Change in Mode=∣80−50∣=30Change\, in\, Mode = |80 -50| =30The statistical measure that would change the most if one of the80's is replaced with a50 is the mode, as it changes by30.

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