QuestionGiven a stem-and-leaf plot, find the mean, median, standard deviation, range, and data shape (right, left, symmetric).

Studdy Solution

STEP 1

Assumptions1. The given data is in a stem-and-leaf plot. . The key indicates that the stem represents tens and the leaves represent ones.
3. The data is a sample, not the entire population.

STEP 2

First, we need to convert the stem-and-leaf plot into a list of numbers. This can be done by combining each stem with each of its leaves.

STEP 3

The converted data is as follows19,33,36,42,42,43,45,50,50,53,54,55,56,62,66,67,69

STEP 4

To find the mean, we sum all the numbers and divide by the count of numbers. The formula for the mean is $Mean = \frac{Sum\, of\, all\, numbers}{Count\, of\, numbers}$

STEP 5

Plug in the values to calculate the mean.
$Mean = \frac{19 +33 +36 +42 +42 +43 +45 +50 +50 +53 +54 +55 +56 +62 +66 +67 +69}{17}$

STEP 6

Calculate the mean.
$Mean = \frac{846}{17} =49.765$

STEP 7

To find the median, we need to sort the numbers in ascending order and find the middle number. If there is an even number of numbers, the median is the average of the two middle numbers.

STEP 8

The numbers are already sorted in ascending order, and there are17 numbers, so the median is theth number.
$Median =50$

STEP 9

To find the standard deviation, we need to find the square root of the variance. The formula for the variance is $Variance = \frac{\sum (x_i - Mean)^2}{n -}$ where $x_i$ are the data points, $Mean$ is the mean, and $n$ is the number of data points.

STEP 10

First, calculate the sum of the squared differences from the mean.
$\sum (x_i - Mean)^2 = (19 -49.765)^2 + (33 -49.765)^2 + \cdots + (69 -49.765)^2$

STEP 11

Calculate the variance.
Variance = \frac{\sum (x_i - Mean)^}{n -} = \frac{\sum (x_i - Mean)^}{16}

STEP 12

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

STEP 13

To find the range, subtract the smallest number from the largest number.
$Range = Largest\, number - Smallest\, number$

STEP 14

Plug in the values to calculate the range.
$Range =69 -19$

STEP 15

Calculate the range.
$Range =69 -19 =50$

STEP 16

To determine the shape of the data, we look at the skewness. If the mean is greater than the median, the data is right-skewed. If the mean is less than the median, the data is left-skewed. If the mean is equal to the median, the data is symmetric.

STEP 17

Compare the mean and the median.
$Mean =49.765, Median =50$ Since the mean is less than the median, the data is left-skewed.

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