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

Studdy Solution

STEP 1

Assumptions1. The data set is $\{6,3,8,6,4,13\}$ . The number of data points, $n$ , is6

STEP 2

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

STEP 3

Plug in the values for $x_i$ and $n$ to calculate the mean.
$Mean = \frac{6 +3 +8 +6 + +13}{6}$

STEP 4

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

STEP 5

To compute the median, we first sort the data set in ascending order and then find the middle value. If there is an even number of data points, the median is the average of the two middle values.
The sorted data set is $\{3,4,,,8,13\}$

STEP 6

Calculate the median.
Since we have an even number of data points, the median is the average of the third and fourth values.
$Median = \frac{6 +6}{2} =6$

STEP 7

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

STEP 8

The range is the difference between the highest and lowest values in the data set.
$Range = Max(x_i) - Min(x_i)$

STEP 9

Plug in the values for $Max(x_i)$ and $Min(x_i)$ to calculate the range.
$Range =13 -3 =$

STEP 10

To compute the variance, we first find the difference between each data point and the mean, square these differences, add them up, and then divide by the number of data points minus.
$Variance = \frac{\sum_{i=}^{n} (x_i - Mean)^2}{n-}$

STEP 11

Plug in the values for $x_i$ , $Mean$ , and $n$ to calculate the variance.
Variance = \frac{(6-6.67)^ + (3-6.67)^ + (8-6.67)^ + (6-6.67)^ + (4-6.67)^ + (13-6.67)^}{6-}

STEP 12

Calculate the variance.
$Variance =12.666$

STEP 13

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

STEP 14

Plug in the value for the variance to calculate the standard deviation.
$Standard\, Deviation = \sqrt{12.666}$

STEP 15

Calculate the standard deviation.
$Standard\, Deviation =3.559$

STEP 16

The coefficient of variation is the standard deviation divided by the mean, usually expressed as a percentage.
$Coefficient\, of\, Variation = \frac{Standard\, Deviation}{Mean} \times100\%$

STEP 17

Plug in the values for the standard deviation and the mean to calculate the coefficient of variation.
$Coefficient\, of\, Variation = \frac{3.559}{6.67} \times100\%$

STEP 18

Calculate the coefficient of variation.
$Coefficient\, of\, Variation =53.34\%$

STEP 19

The Z scores are calculated by subtracting the mean from each data point and then dividing by the standard deviation.
$Z = \frac{x_i - Mean}{Standard\, Deviation}$

STEP 20

Calculate the Z scores for each data point.
$Z = \frac{6 -6.67}{3.559}, \frac{3 -6.67}{3.559}, \frac{8 -6.67}{3.559}, \frac{6 -6.67}{3.559}, \frac{4 -6.67}{3.559}, \frac{13 -6.67}{3.559}$

STEP 21

A Z score of greater than3 or less than -3 is considered an outlier. In this case, none of the Z scores exceed these values, so there are no outliers.

STEP 22

The shape of the data set can be described by its skewness and kurtosis. Since these are not provided, we can make a qualitative assessment. The data set appears to be slightly right-skewed, as it has a longer tail on the right side.

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