QuestionGiven data: 6, 3, 8, 6, 4, 13 (n=6). Find mean, median, mode, range, variance, std. dev., Z scores, and data 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 calculate the mean, we add up all the numbers in the data set and then divide by the number of data points.
$Mean = \frac{\sum_{i=1}^{n} x_i}{n}$

STEP 3

Now, plug in the given values for the data set and $n$ to calculate the mean.
$Mean = \frac{6 +3 +8 +6 + +13}{6}$

STEP 4

Calculate the mean.
$Mean = \frac{40}{6} =6.6667$

STEP 5

To calculate the median, we first need to sort the data set in ascending order.
The sorted data set is $\{3,4,,,8,13\}$ .

STEP 6

Since the number of data points is even, the median is the average of the two middle numbers.
$Median = \frac{6 +6}{2}$

STEP 7

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

STEP 8

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

STEP 9

To calculate the range, we subtract the smallest number in the data set from the largest number.
$Range = max(x_i) - min(x_i)$

STEP 10

Now, plug in the given values for the data set to calculate the range.
$Range =13 -3$

STEP 11

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

STEP 12

To calculate the variance, we first calculate the squared difference from the mean for each data point, then take the average of these squared differences.
$Variance = \frac{\sum_{i=}^{n} (x_i - Mean)^2}{n}$

STEP 13

Now, plug in the given values for the data set and the mean to calculate the variance.
$Variance = \frac{(6-6.6667)^2 + (3-6.6667)^2 + (8-6.6667)^2 + (6-6.6667)^2 + (-6.6667)^2 + (13-6.6667)^2}{6}$

STEP 14

Calculate the variance.
$Variance =9.5556$

STEP 15

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

STEP 16

Now, plug in the variance to calculate the standard deviation.
$Standard\, deviation = \sqrt{9.5556}$

STEP 17

Calculate the standard deviation.
$Standard\, deviation =3.0912$

STEP 18

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

STEP 19

Now, plug in the standard deviation and the mean to calculate the coefficient of variation.
$Coefficient\, of\, variation = \frac{3.091}{6.6667} \\times100\%$

STEP 20

Calculate the coefficient of variation.
$Coefficient\, of\, variation =46.34\%$

STEP 21

The Z score for a data point is the number of standard deviations it is from the mean.
$Z\, score = \frac{x_i - Mean}{Standard\, deviation}$

STEP 22

Now, plug in the given values for the data set, the mean, and the standard deviation to calculate the Z scores.

STEP 23

Calculate the Z scores.
For $x_i =6$ , $Z\, score = \frac{6 -6.6667}{3.091} = -0.2165$
For $x_i =3$ , $Z\, score = \frac{3 -6.6667}{3.091} = -1.187$
For $x_i =8$ , $Z\, score = \frac{8 -6.6667}{3.091} =0.4313$
For $x_i =$ , $Z\, score = \frac{ -6.6667}{3.091} = -0.8629$
For $x_i =13$ , $Z\, score = \frac{13 -6.6667}{3.091} =.0485$

STEP 24

Outliers are typically defined as data points with a Z score of less than -3 or greater than3. In this case, there are no outliers.

STEP 25

The shape of the data set can be described by its skewness and kurtosis.Since the mean (.6667) is greater than the median (), the data set is slightly skewed to the right.
Since there are no outliers, the data set has a low kurtosis.

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