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QuestionGiven data: 6, 3, 8, 6, 4, 13. Find mean, median, mode, range, variance, standard deviation, Z scores, and CV. Are there outliers?

Studdy Solution

STEP 1

Assumptions1. The data set is {6,3,8,6,4,13}\{6,3,8,6,4,13\} . The number of data points, nn, is63. The coefficient of variation is calculated as the ratio of the standard deviation to the mean, expressed as a percentage4. Outliers are typically defined as data points that have a ZZ score greater than3 or less than -3

STEP 2

First, we need to compute the mean of the data set. The mean is the sum of all data points divided by the number of data points.
Mean=Sum of all data pointsNumber of data pointsMean = \frac{Sum\, of\, all\, data\, points}{Number\, of\, data\, points}

STEP 3

Now, plug in the given values for the data points and the number of data points to calculate the mean.
Mean=6+3+8+6++136Mean = \frac{6 +3 +8 +6 + +13}{6}

STEP 4

Calculate the mean.
Mean=406=6.67Mean = \frac{40}{6} =6.67

STEP 5

Next, we need to compute the median of the data set. The median is the middle value when the data points are arranged in ascending order. If there is an even number of data points, the median is the average of the two middle numbers.
Median=th value+7th value2Median = \frac{th\, value +7th\, value}{2}

STEP 6

Arrange the data points in ascending order and find the median.
Median=6+62Median = \frac{6 +6}{2}

STEP 7

Calculate the median.
Median=122=6Median = \frac{12}{2} =6

STEP 8

The mode is the value that appears most frequently in a data set. In this case, the mode is6.

STEP 9

Next, we need to compute the range of the data set. The range is the difference between the highest and lowest values.
Range=Highest value−Lowest valueRange = Highest\, value - Lowest\, value

STEP 10

Plug in the highest and lowest values to calculate the range.
Range=13−3Range =13 -3

STEP 11

Calculate the range.
Range=13−3=10Range =13 -3 =10

STEP 12

Next, we need to compute the variance of the data set. The variance is the average of the squared differences from the mean.
Variance=∑(xi−Mean)2nVariance = \frac{\sum{(x_i - Mean)^2}}{n}

STEP 13

Plug in the values for each data point, the mean, and the number of data points to calculate the variance.
Variance=(6−6.67)2+(3−6.67)2+(8−6.67)2+(6−6.67)2+(−6.67)2+(13−6.67)26Variance = \frac{(6-6.67)^2 + (3-6.67)^2 + (8-6.67)^2 + (6-6.67)^2 + (-6.67)^2 + (13-6.67)^2}{6}

STEP 14

Calculate the variance.
Variance=0.4489+13.4489+.7889+0.4489+7.1124+40.11246=10.5602Variance = \frac{0.4489 +13.4489 +.7889 +0.4489 +7.1124 +40.1124}{6} =10.5602

STEP 15

The standard deviation is the square root of the variance.
Standard Deviation=VarianceStandard\, Deviation = \sqrt{Variance}

STEP 16

Plug in the variance to calculate the standard deviation.
Standard Deviation=10.5602Standard\, Deviation = \sqrt{10.5602}

STEP 17

Calculate the standard deviation.
Standard Deviation=10.5602=3.25Standard\, Deviation = \sqrt{10.5602} =3.25

STEP 18

The coefficient of variation is the ratio of the standard deviation to the mean, expressed as a percentage.
Coefficient of Variation=Standard DeviationMean×100%Coefficient\, of\, Variation = \frac{Standard\, Deviation}{Mean} \times100\%

STEP 19

Plug in the standard deviation and the mean to calculate the coefficient of variation.
Coefficient of Variation=3.256.67×100%Coefficient\, of\, Variation = \frac{3.25}{6.67} \times100\%

STEP 20

Calculate the coefficient of variation.
Coefficient of Variation=3.256.67×100%=48.73%Coefficient\, of\, Variation = \frac{3.25}{6.67} \times100\% =48.73\%

STEP 21

The ZZ score for a data point is the number of standard deviations it is from the mean.
Z Score=xi−MeanStandard DeviationZ\, Score = \frac{x_i - Mean}{Standard\, Deviation}

STEP 22

Calculate the ZZ score for each data point.
Z Scores=6−6.67.25,−6.67.25,8−6.67.25,6−6.67.25,4−6.67.25,13−6.67.25Z\, Scores = \frac{6 -6.67}{.25}, \frac{ -6.67}{.25}, \frac{8 -6.67}{.25}, \frac{6 -6.67}{.25}, \frac{4 -6.67}{.25}, \frac{13 -6.67}{.25}

STEP 23

Calculate the ZZ scores.
Z Scores=−0.21,−1.13,0.41,−0.21,−0.82,1.95Z\, Scores = -0.21, -1.13,0.41, -0.21, -0.82,1.95

STEP 24

There are no outliers in the data set, as all ZZ scores are between -3 and3.

STEP 25

The shape of the data set can be described as slightly right-skewed, as the mean (.67) is greater than the median ().

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