1. A class took a test. The students' got the following scores.

94 85 74 85 77 100 85 95 98 95

Find the mean, median, mode, upper quartile, lower quartile, interquartile range, and standard deviation. Draw the stem and leaf plot, the box and whisker plot, the frequency distribution and the frequency histogram. Are there any outliers?

First let us draw the stem and leaf plot

This makes it easy to see that the mode, the most common score is an 85 since there are three of those scores and all of the other scores have frequencies of either one or tow.

It will also make it easier to rank the scores

This will make it easier to find the median and the quartiles. There are as many scores above the median as below. Since there are ten scores, and ten is an even number, we can divide the scores into two equal groups with no scores left over. To find the median, we divide the scores up into the upper five and the lower five.

We see that the median will be halfway between 94 and 85. To find the number that is halfway between these two numbers, we add them up and divide the sum by 2.

The upper quartile is the median of the upper half of the scores, and the lower quartile is the median of the lower half of the scores. There are five scores in both of these halves, and since 5 is an odd number, there will be a numbere that is the middle score. Since half of five is two and a half the quartiles will be between the upper two score and lower two scores in each of the halves.

We see that Q1 = 85 and Q3 = 85. Therefore the interquartile range IQR = 95 - 85 = 10.

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With this, we can make our box and whisker plot

An outlier is more than 1.5(IQR) either above the upper quartile or below the lower quartile. 1.5(IQR) = 15. An outlier would either have to be more than 85 - 15 = 70 or 95 + 15 = 110. The 74 is getting close to being an outlier but is not. There are no outliers.

The stem and leaf plot is also helpful for getting the frequency distribution.

 

From the frequency distribution we can graph the frequency histogram.

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The frequency distribution is helpful for computing the mean and standard deviation. First we compute the mean. Add up all the scores and divide by the nuber of scores.

The mean is 888/10 = 88.8

To get the standard deviation, we first compute the deviations by subtracting the mean from all of the scores.

Then we need to square the deviations.

To total the squares of the deviations, we need to remember that some of these scores have frequencies. We multiply the squares of the deviations by the frequencies of the scores and total up the products.

Since the students taking the test is a sample, we use the sample standard deviation. To get the sample variance, we divide the sum of the squares of the deviations by 9, which is 1 less than the sample size.

and the sample variance is 79.5111. . .

To get the sample standard deviation, we take the square root of the sample variance.

and the sample standard deviation is approximately 8.9169.

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