14. A class takes a quiz and gets the following scores.

 

22, 23, 25, 33, 33, 33, 38, 41, 44, 44, 49

 

Make up a frequency distribution. Find the mean, median, mode, upper and lower quartiles, interquartile range, and standard deviation. Make up a stem and leaf plot and a box and whisker plot.

The frequency distribution looks like

The mean is the sum of the scores divided by the number of scores.We can use the frequency distribution to find the mean. Since there are 3 33s they add up to 33 x 3. The sum of the scores is the sum of the product of each different score times its frequency. The total number of scores is the sum of the frequencies

We divide the sum of the scores which is 385 by the number of scores which is 11.

385/11 = 35

so the mean is 35. We are very lucky that the mean comes out even. It quite often doesn't.

To find the median, since there are 11 scores and 11 is an odd number, there will be a middle score which will separate the top 5 score from the bottom 5 scores.

We see that the middle number is one of the 33s, so the median is 33.

The mode is the most common score, which we see from the frequency distribution is also 33.

For the upper quartile, find the median of the scores which are above the median, and for the lower quartile, find the median of the scores which are below the median. Since there are 5 score above the median and 5 scores below the median, and since 5 is an odd number, there will be a score in the middle which will divide the upper 2 score from the lower 2 scores for each of these groups of 5 scores.

The upper quartile is 44, and the lower quartile is 25 so the Inter Quartile Range is

44 - 25 = 19

For the standard deviation, we first compute the deviations from the mean.

Then we square all the deviations

To find the mean of the squares of the deviations, we must take into account the frequencies of the sores, and add up the products of the squares of the deviations times the frequencies.

To get the mean of the squares of the deviations, we divide their sum which is 828 by 11 which is the number of scores.

The mean of the squares of the deviations is the variance. The standard deviation is the square root of the variance.

For the stem and leaf plot, look at the scores

22, 23, 25, 33, 33, 33, 38, 41, 44, 44, 49.

The numbers in the ten's places, which are the stems, are 2, 3, and 4

For the box and whisker plot, recall

that the upper and lower quartile were 44 and 25 respectively. Plot these, the median, the max and min on a number line. The box and whisker plot looks like

An outlier is a score that is either 1.5 IQR above the upper quartile or 1.5 IQR below the lower quartile. Since the IQR is 19, 1.5 IQR is 28.5. 28.5 above the upper quartile of 44 would be 72.5, and we do not have any scores that big. 28.5 below the lower quartile of 25 would be -3.5, and we don't have any scores that low.