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QuestionGiven data: 6,3,8,6,4,136, 3, 8, 6, 4, 13. Find mean, median, mode, range, variance, std. dev., Z scores, and describe shape.

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, is6

STEP 2

To compute the mean, we sum all the numbers in the data set and divide by the number of data points.
Mean=∑i=1nxinMean = \frac{\sum_{i=1}^{n} x_i}{n}

STEP 3

Plug in the values from the data set and the number of data points into the formula 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

To compute the median, we first need to arrange the data set in ascending order.
{3,4,,,8,13}\{3,4,,,8,13\}

STEP 6

Since the number of data points is even, the median is the average of the two middle numbers.
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 number that appears most frequently in the data set. In this case, it's6.

STEP 9

To compute the range, we subtract the smallest number in the data set from the largest.
Range=Max−MinRange = Max - Min

STEP 10

Plug in the maximum and minimum values from the data set into the formula to calculate the range.
Range=13−3Range =13 -3

STEP 11

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

STEP 12

The variance is given as 2=12.666^2 =12.666.

STEP 13

To compute the standard deviation, we take the square root of the variance.
=2 = \sqrt{^2}

STEP 14

Plug in the value for the variance into the formula to calculate the standard deviation.
=12.666 = \sqrt{12.666}

STEP 15

Calculate the standard deviation.
=12.666=3.56 = \sqrt{12.666} =3.56

STEP 16

To compute the Z scores, we subtract the mean from each data point and divide by the standard deviation.
Z=x−MeanZ = \frac{x - Mean}{}

STEP 17

Calculate the Z scores for each data point in the data set.
Z=6−6.673.56,3−6.673.56,−6.673.56,6−6.673.56,4−6.673.56,13−6.673.56Z = \frac{6 -6.67}{3.56}, \frac{3 -6.67}{3.56}, \frac{ -6.67}{3.56}, \frac{6 -6.67}{3.56}, \frac{4 -6.67}{3.56}, \frac{13 -6.67}{3.56}

STEP 18

Calculate the Z scores.
Z=−0.,−.03,0.37,−0.,−0.75,.78Z = -0., -.03,0.37, -0., -0.75,.78

STEP 19

A Z score greater than or less than - is considered an outlier. In this data set, there are no outliers.

STEP 20

The shape of the data set can be described by its skewness and kurtosis. Since the mean (6.67) is slightly greater than the median (6), the data set is slightly skewed to the right. The presence of a single mode (6) suggests that the data set has a unimodal distribution.

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