1. Some members of a class took a test. The scores were

summarized in the following frequency distribution. Find the mean, median, mode, upper and lower quartiles, the interquartile range, and standard deviation of the scores, and draw the histogram.

x

|

f




90

|

2

80

|

3

70

|

1

60

|

4

To find the mean, we add up all the scores and divide by the number of scores. The two 90s will add up to 2x90 etc

x

|

f

xf




90

|

2

180

80

|

3

240

70

|

1

70

60

|

4

240

totals

10

730

To get the mean, we divide the total of the score by the number of scores, which is the sum of the frequencies

mean = 730/10 = 73

The median is the middle score. To find it, rank the scores and divide them into two equal groups, the upper half of the scores and the lower half.

The mnidpoint is between 70 and 80. The nuimber which is halfway between 70 and 80 is 75.

The mode is the most cmmon score. That is 60.

The upper quartile is the median of the upper half of the scores ånd the lower quartile is the median of the lower half of the scores.There are five scores in the upper half of the scores and five in the lower half. Five is an odd number so when you divide the scores in to two equal groups, there will be one in the middle.

Half of five is two and a half, so when we divide the scores into two equal groups we get a lower two, an upper two, and one score in the middle. That middle score is the median of the group. So we see that the upper quartile is at 80, and the lower quartile is at 60.

The interquartile range is then

80 - 60 = 20

To get the standard deviation, we first find the deviations of all the scores. Recall that the mean was 73.

x

|

f

dev




90

|

2

17

80

|

3

7

70

|

1

-3

60

|

4

-13

Next square all the deviations.

x

|

f

dev

dev sq




90

|

2

17

289

80

|

3

7

49

70

|

1

-3

9

60

|

4

-13

169

To total up the squares of the deviations, we need to take the frequencies into account.

x

|

f

dev

dev sq

(dev sq)f




90

|

2

17

289

578

80

|

3

7

49

147

70

|

1

-3

9

9

60

|

4

-13

169

676

total

10

1410

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

variance = 1410/10 = 141

The standard deviation is the square root of the variance or

11.874342087

This is the population standard deviation. To get the sample standard deviation, instead of dividing by the number of scores, divide by one less than the number of scores.

1410/9 = 156.666...

with 6's going on forever. This is the sample variance. The sample standard deviation is the square root of this

12.516656

After seeing how to compute the standard deviations, you may well prefer to use a calculator. Different calculators behave differently, and even if I were to discuss your calculator, it might not be too long before it becomes obsolete. If you have the book that came with your calculator, consult it. You can check that you are doing it correctly by seeing if you get these answers.

The histogram looks like