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Question Which measure of central tendency is most affected by the extreme value in the given half marathon pace data?
{6.7,7.1,7.5,7.6,3.3,7.4,7.2,7.4,7.1,7.4,6.8,6.9,7.3}\{6.7, 7.1, 7.5, 7.6, 3.3, 7.4, 7.2, 7.4, 7.1, 7.4, 6.8, 6.9, 7.3\}

Studdy Solution

STEP 1

Assumptions
1. We are given a set of pace values for a half marathon measured in miles per hour (mph).
2. We need to determine which measure of central tendency (mean, mode, median) is most affected by the extreme value in the data set.
3. The extreme value in the data set is 3.3 mph, which is noticeably lower than the other values.

STEP 2

First, let's list all the pace values in ascending order to help us identify the median and to calculate the mean.
3.3,6.7,6.8,6.9,7.1,7.1,7.2,7.3,7.4,7.4,7.4,7.4,7.5,7.63.3, 6.7, 6.8, 6.9, 7.1, 7.1, 7.2, 7.3, 7.4, 7.4, 7.4, 7.4, 7.5, 7.6

STEP 3

Now, let's calculate the mean (average) pace. The mean is calculated by adding all the values together and then dividing by the number of values.
Mean=∑Pace valuesNumber of valuesMean = \frac{\sum \text{Pace values}}{\text{Number of values}}

STEP 4

Add all the pace values together.
Sum=3.3+6.7+6.8+6.9+7.1+7.1+7.2+7.3+7.4+7.4+7.4+7.4+7.5+7.6Sum = 3.3 + 6.7 + 6.8 + 6.9 + 7.1 + 7.1 + 7.2 + 7.3 + 7.4 + 7.4 + 7.4 + 7.4 + 7.5 + 7.6

STEP 5

Calculate the sum of the pace values.
Sum=98.1Sum = 98.1

STEP 6

Count the number of pace values.
Number of values=14Number\, of\, values = 14

STEP 7

Calculate the mean pace.
Mean=98.114Mean = \frac{98.1}{14}

STEP 8

Compute the mean.
Mean≈7.007Mean \approx 7.007

STEP 9

To find the mode, we look for the value that appears most frequently in the data set.

STEP 10

Identify the mode by observing the frequency of each value.
Mode=7.4 (since it appears 4 times)Mode = 7.4 \text{ (since it appears 4 times)}

STEP 11

To find the median, we locate the middle value in the ordered list. If there is an even number of values, the median is the average of the two middle numbers.

STEP 12

Since there are 14 values, the median will be the average of the 7th and 8th values in the ordered list.

STEP 13

Identify the 7th and 8th values in the ordered list.
7th value=7.2,8th value=7.37th\, value = 7.2, 8th\, value = 7.3

STEP 14

Calculate the median.
Median=7.2+7.32Median = \frac{7.2 + 7.3}{2}

STEP 15

Compute the median.
Median=7.25Median = 7.25

STEP 16

Now, let's analyze which measure of central tendency is most affected by the extreme value of 3.3 mph.

STEP 17

The mean is sensitive to all values in the data set, including extreme values.

STEP 18

The mode is not affected by the extreme value since it only depends on the frequency of occurrence.

STEP 19

The median is less affected by extreme values because it is the middle value of an ordered list.

STEP 20

Given that the mean is the only measure that takes into account the magnitude of every value, it is the measure most affected by the extreme value.
The measure of central tendency most affected by the extreme value is the mean (F).

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