The math grades on the final exam varied greatly. Using the scores below, how many scores were within one standard deviation of the mean? How many scores were within two standard deviations of the mean?

99 34 86 57 73 85 91 93 46 96 88 79 68 85 89

Find the mean first = sum of scores/number of scores

Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.

Standard deviation = square root of variance

I'll let you do the calculations.

To find the number of scores within one standard deviation of the mean, we first need to calculate the mean and the standard deviation.

Step 1: Find the mean (average):
Add up all the scores and divide by the total number of scores:

99 + 34 + 86 + 57 + 73 + 85 + 91 + 93 + 46 + 96 + 88 + 79 + 68 + 85 + 89 = 1253

Mean (Average) = 1253 / 15 = 83.53 (rounded to two decimal places)

Step 2: Find the standard deviation:
To find the standard deviation, we need to calculate the deviation of each score from the mean and then find the average of the squared deviations. Finally, we take the square root of this average.

Subtract the mean from each score and square the result:

(99 - 83.53)^2 = 239.23
(34 - 83.53)^2 = 2432.90
(86 - 83.53)^2 = 0.06
(57 - 83.53)^2 = 711.44
(73 - 83.53)^2 = 110.29
(85 - 83.53)^2 = 2.13
(91 - 83.53)^2 = 54.21
(93 - 83.53)^2 = 89.82
(46 - 83.53)^2 = 1394.89
(96 - 83.53)^2 = 155.93
(88 - 83.53)^2 = 19.93
(79 - 83.53)^2 = 21.37
(68 - 83.53)^2 = 239.65
(85 - 83.53)^2 = 2.13
(89 - 83.53)^2 = 29.61

Summing up all the squared deviations:

239.23 + 2432.90 + 0.06 + 711.44 + 110.29 + 2.13 + 54.21 + 89.82 + 1394.89 + 155.93 + 19.93 + 21.37 + 239.65 + 2.13 + 29.61 = 5773.47

Divide the sum of squared deviations by the total number of scores (15 in this case):

Standard Deviation = √(5773.47 / 15) = 17.90 (rounded to two decimal places)

Step 3: Calculate the one standard deviation range:
One standard deviation range is calculated by adding and subtracting one standard deviation from the mean.

Lower Limit = Mean - (Standard Deviation * 1)
Upper Limit = Mean + (Standard Deviation * 1)

Lower Limit = 83.53 - (17.90 * 1) = 65.63 (rounded to two decimal places)
Upper Limit = 83.53 + (17.90 * 1) = 101.43 (rounded to two decimal places)

Any score within this range (65.63 to 101.43) is within one standard deviation of the mean.

Step 4: Count the scores within one standard deviation of the mean:
By comparing each score with the one standard deviation range, we can count the number of scores within this range.

Scores within one standard deviation: 99, 86, 91, 93, 88, 79, 85, 89

There are 8 scores within one standard deviation of the mean.

Step 5: Calculate the two standard deviations range:
Similar to step 3, the two standard deviations range is calculated by adding and subtracting two times the standard deviation from the mean.

Lower Limit = Mean - (Standard Deviation * 2)
Upper Limit = Mean + (Standard Deviation * 2)

Lower Limit = 83.53 - (17.90 * 2) = 47.73 (rounded to two decimal places)
Upper Limit = 83.53 + (17.90 * 2) = 119.33 (rounded to two decimal places)

Any score within this range (47.73 to 119.33) is within two standard deviations of the mean.

Step 6: Count the scores within two standard deviations of the mean:
By comparing each score with the two standard deviations range, we can count the number of scores within this range.

Scores within two standard deviations: 99, 34, 86, 57, 73, 85, 91, 93, 46, 96, 88, 79, 68, 85, 89

There are 15 scores within two standard deviations of the mean.

To summarize:

Number of scores within one standard deviation of the mean: 8
Number of scores within two standard deviations of the mean: 15

To find how many scores were within one standard deviation of the mean, we first need to find the mean and standard deviation of the scores. Let's go through the steps:

1. Find the Mean:
The mean (average) can be found by summing up all the scores and dividing it by the total number of scores. In this case, we have 15 scores.

Mean = (99 + 34 + 86 + 57 + 73 + 85 + 91 + 93 + 46 + 96 + 88 + 79 + 68 + 85 + 89) / 15

Mean = 1236 / 15

Mean = 82.4

So, the mean of the scores is 82.4.

2. Find the Standard Deviation:
The standard deviation measures how much the scores deviate from the mean. Calculating it requires several steps:

a. Find the differences between each score and the mean.
Subtract the mean (82.4) from each of the scores individually.

99 - 82.4 = 16.6
34 - 82.4 = -48.4
86 - 82.4 = 3.6
57 - 82.4 = -25.4
73 - 82.4 = -9.4
85 - 82.4 = 2.6
91 - 82.4 = 8.6
93 - 82.4 = 10.6
46 - 82.4 = -36.4
96 - 82.4 = 13.6
88 - 82.4 = 5.6
79 - 82.4 = -3.4
68 - 82.4 = -14.4
85 - 82.4 = 2.6
89 - 82.4 = 6.6

b. Square each of the differences.
Square each of the differences calculated in step a. This step ensures that negative differences don't cancel out positive differences.

16.6^2 = 275.56
(-48.4)^2 = 2342.56
3.6^2 = 12.96
(-25.4)^2 = 645.16
(-9.4)^2 = 88.36
2.6^2 = 6.76
8.6^2 = 73.96
10.6^2 = 112.36
(-36.4)^2 = 1324.96
13.6^2 = 184.96
5.6^2 = 31.36
(-3.4)^2 = 11.56
(-14.4)^2 = 207.36
2.6^2 = 6.76
6.6^2 = 43.56

c. Find the average of the squared differences.
Add up all the squared differences from step b and divide the sum by the total number of scores (15).

Sum of squared differences = 275.56 + 2342.56 + 12.96 + 645.16 + 88.36 + 6.76 + 73.96 + 112.36 + 1324.96 + 184.96 + 31.36 + 11.56 + 207.36 + 6.76 + 43.56

Sum of squared differences = 5086.72

Standard Deviation = sqrt(5086.72 / 15)

Standard Deviation ≈ 14.06 (rounded to two decimal places)

3. Find the number of scores within one standard deviation of the mean:
Scores within one standard deviation of the mean are those that fall within a range of one standard deviation above or below the mean.

To calculate this, we add or subtract one standard deviation (14.06) from the mean (82.4):

Lower bound = 82.4 - 14.06 = 68.34
Upper bound = 82.4 + 14.06 = 96.46

Count the scores that fall within this range:

68, 73, 79, 85, 86, 88, 89, 91, 93, 96

There are 10 scores within one standard deviation of the mean.

4. Find the number of scores within two standard deviations of the mean:
Scores within two standard deviations of the mean are those that fall within a range of two standard deviations above or below the mean.

To calculate this, we add or subtract two standard deviations (14.06 * 2) from the mean (82.4):

Lower bound = 82.4 - (14.06 * 2) = 54.28
Upper bound = 82.4 + (14.06 * 2) = 110.52

Count the scores that fall within this range:

34, 46, 57, 68, 73, 79, 85, 86, 88, 89, 91, 93, 96

There are 13 scores within two standard deviations of the mean.

Therefore, there were 10 scores within one standard deviation of the mean and 13 scores within two standard deviations of the mean.