QuestionGiven data: $n=6$ , values: 6, 3, 8, 6, 4, 13.
a. Find mean, median, mode. b. Find range, variance, standard deviation, coefficient of variation. c. Compute Z scores and identify outliers. d. Describe data shape. $S^{<}=12.666$ , $S=3.559$ , $\mathrm{CV}=\square$ .

Studdy Solution

STEP 1

Assumptions1. The data set is $\{6,3,8,6,4,13\}$ . The sample size is $n=6$
3. The sum of the data set is $^{<}=12.666$
4. The standard deviation is $=3.559$
5. The coefficient of variation is to be calculated

STEP 2

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

STEP 3

Now, plug in the given values for the sum of the data and the number of data points to calculate the mean.
$Mean = \frac{12.666}{6}$

STEP 4

Calculate the mean.
$Mean = \frac{12.666}{6} =2.111$

STEP 5

Next, we need to calculate the median of the data set. The median is the middle value when the data is arranged in ascending order. If there is an even number of data points, the median is the average of the two middle numbers.
The data set in ascending order is $\{3,4,,,8,13\}$ So, the median is the average of the third and fourth data points.
$Median = \frac{ +}{2}$

STEP 6

Calculate the median.
$Median = \frac{6 +6}{2} =6$

STEP 7

The mode of a data set is the value that appears most frequently. In this case, the mode is6 as it appears twice and all other numbers appear only once.
$Mode =6$

STEP 8

The range of a data set is the difference between the highest and lowest values. In this case, the range is $13 -3$ .
$Range =13 -3$

STEP 9

Calculate the range.
$Range =13 -3 =$

STEP 10

The variance of a data set is the average of the squared differences from the mean. However, we are given the standard deviation, which is the square root of the variance. So, we can find the variance by squaring the standard deviation.
$Variance = (Standard\, deviation)^2$

STEP 11

Now, plug in the given value for the standard deviation to calculate the variance.
Variance = (3.559)^

STEP 12

Calculate the variance.
$Variance = (.559)^2 =12.665$

STEP 13

The standard deviation is given as $=3.559$ .

STEP 14

The coefficient of variation (CV) is a measure of relative variability. It is the ratio of the standard deviation to the mean, often expressed as a percentage.
$CV = \frac{Standard\, deviation}{Mean} \times100\%$

STEP 15

Now, plug in the given values for the standard deviation and the mean to calculate the CV.
$CV = \frac{3.559}{2.111} \times100\%$

STEP 16

Calculate the CV.
$CV = \frac{3.559}{2.111} \times100\% =168.5\%$

STEP 17

Z-scores are a measure of how many standard deviations an element is from the mean. The Z-score for an element $x$ is calculated as follows $Z = \frac{x - Mean}{Standard\, deviation}$

STEP 18

Now, calculate the Z-scores for all the data points. A Z-score greater than3 or less than -3 is considered an outlier.

STEP 19

The shape of the data set can be described by its skewness and kurtosis. However, with such a small data set, it's hard to make any definitive conclusions about its shape.
The mean is less than the median, which suggests the data might be slightly skewed to the left. However, the difference is very small, so it's also possible that the data is approximately symmetric.

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Studdy Solution

STEP 1

Assumptions1. The data set is {6,3,8,6,4,13}\{6,3,8,6,4,13\}{6,3,8,6,4,13} . The sample size is n=6n=6n=63. The sum of the data set is <=12.666^{<}=12.666<=12.6664. The standard deviation is =3.559=3.559=3.5595. The coefficient of variation is to be calculated

STEP 2

First, we need to calculate the mean of the data set. The mean is the sum of the data divided by the number of data points.Mean=Sum of dataNumber of data pointsMean = \frac{Sum\, of\, data}{Number\, of\, data\, points}Mean=NumberofdatapointsSumofdata​

STEP 3

Now, plug in the given values for the sum of the data and the number of data points to calculate the mean.Mean=12.6666Mean = \frac{12.666}{6}Mean=612.666​

STEP 4

Calculate the mean.Mean=12.6666=2.111Mean = \frac{12.666}{6} =2.111Mean=612.666​=2.111

STEP 5

STEP 6

Calculate the median.Median=6+62=6Median = \frac{6 +6}{2} =6Median=26+6​=6

STEP 7

The mode of a data set is the value that appears most frequently. In this case, the mode is6 as it appears twice and all other numbers appear only once.Mode=6Mode =6Mode=6

STEP 8

The range of a data set is the difference between the highest and lowest values. In this case, the range is 13−313 -313−3.Range=13−3Range =13 -3Range=13−3

STEP 9

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

STEP 10

STEP 11

Now, plug in the given value for the standard deviation to calculate the variance.Variance = (3.559)^

STEP 12

Calculate the variance.Variance=(.559)2=12.665Variance = (.559)^2 =12.665Variance=(.559)2=12.665

STEP 13

The standard deviation is given as =3.559=3.559=3.559.

STEP 14

The coefficient of variation (CV) is a measure of relative variability. It is the ratio of the standard deviation to the mean, often expressed as a percentage.CV=Standard deviationMean×100%CV = \frac{Standard\, deviation}{Mean} \times100\%CV=MeanStandarddeviation​×100%

STEP 15

Now, plug in the given values for the standard deviation and the mean to calculate the CV.CV=3.5592.111×100%CV = \frac{3.559}{2.111} \times100\%CV=2.1113.559​×100%

STEP 16

Calculate the CV.CV=3.5592.111×100%=168.5%CV = \frac{3.559}{2.111} \times100\% =168.5\%CV=2.1113.559​×100%=168.5%

STEP 17

Z-scores are a measure of how many standard deviations an element is from the mean. The Z-score for an element xxx is calculated as followsZ=x−MeanStandard deviationZ = \frac{x - Mean}{Standard\, deviation}Z=Standarddeviationx−Mean​

STEP 18

Now, calculate the Z-scores for all the data points. A Z-score greater than3 or less than -3 is considered an outlier.

STEP 19

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Assumptions1. The data set is $\{6,3,8,6,4,13\}$ . The sample size is $n=6$
3. The sum of the data set is $^{<}=12.666$
4. The standard deviation is $=3.559$
5. The coefficient of variation is to be calculated

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

Now, plug in the given values for the sum of the data and the number of data points to calculate the mean.
$Mean = \frac{12.666}{6}$

Calculate the mean.
$Mean = \frac{12.666}{6} =2.111$

Calculate the median.
$Median = \frac{6 +6}{2} =6$

The mode of a data set is the value that appears most frequently. In this case, the mode is6 as it appears twice and all other numbers appear only once.
$Mode =6$

The range of a data set is the difference between the highest and lowest values. In this case, the range is $13 -3$ .
$Range =13 -3$

Calculate the range.
$Range =13 -3 =$

Now, plug in the given value for the standard deviation to calculate the variance.
Variance = (3.559)^

Calculate the variance.
$Variance = (.559)^2 =12.665$

The standard deviation is given as $=3.559$ .

The coefficient of variation (CV) is a measure of relative variability. It is the ratio of the standard deviation to the mean, often expressed as a percentage.
$CV = \frac{Standard\, deviation}{Mean} \times100\%$

Now, plug in the given values for the standard deviation and the mean to calculate the CV.
$CV = \frac{3.559}{2.111} \times100\%$

Calculate the CV.
$CV = \frac{3.559}{2.111} \times100\% =168.5\%$

Z-scores are a measure of how many standard deviations an element is from the mean. The Z-score for an element $x$ is calculated as follows $Z = \frac{x - Mean}{Standard\, deviation}$