Find the range, standard deviation, and variance for the following sample data:
98, 2, 53, 61, 15, 87, 63, 63, 91, 29, 96, 79, 93, 40, 5, 62
*please show calculations step by step so that i can compare my work. thank you very much. i know that i am off
mary Hornbeck
I can't help you with statistics; however, let me give you something to think about. How many students can we help if we type all of the steps for one student. How many students can we help if YOU do the typing and we find the error. It's a lot more efficient for YOU to do the typing and us to the comparing than the other way around.
Range = highest value - lowest
For variance and standard deviation:
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.
Step 1: Find the range
- The range is the difference between the highest and lowest values in the data set.
- In this case, the highest value is 98 and the lowest value is 2.
- So, the range = 98 - 2 = 96
Step 2: Find the mean
- The mean is the average of all the values in the data set.
- Add up all the values: 98 + 2 + 53 + 61 + 15 + 87 + 63 + 63 + 91 + 29 + 96 + 79 + 93 + 40 + 5 + 62 = 996
- Divide the sum by the number of values in the data set (which is 16 in this case).
- Mean = 996 / 16 = 62.25
Step 3: Find the deviations from the mean
- For each value in the data set, subtract the mean from it.
- Deviations from the mean: 98 - 62.25 = 35.75
2 - 62.25 = -60.25
53 - 62.25 = -9.25
61 - 62.25 = -1.25
15 - 62.25 = -47.25
87 - 62.25 = 24.75
63 - 62.25 = 0.75
63 - 62.25 = 0.75
91 - 62.25 = 28.75
29 - 62.25 = -33.25
96 - 62.25 = 33.75
79 - 62.25 = 16.75
93 - 62.25 = 30.75
40 - 62.25 = -22.25
5 - 62.25 = -57.25
62 - 62.25 = -0.25
Step 4: Find the squared deviations
- Square each deviation from the mean.
- Squared deviations: 35.75^2 = 1,281.56
-60.25^2 = 3,635.06
-9.25^2 = 85.56
-1.25^2 = 1.56
-47.25^2 = 2,229.56
24.75^2 = 612.56
0.75^2 = 0.56
0.75^2 = 0.56
28.75^2 = 828.06
-33.25^2 = 1,106.56
33.75^2 = 1,140.56
16.75^2 = 280.56
30.75^2 = 947.56
-22.25^2 = 494.56
-57.25^2 = 3,280.06
-0.25^2 = 0.06
Step 5: Find the sum of squared deviations
- Add up all the squared deviations: 1,281.56 + 3,635.06 + 85.56 + 1.56 + 2,229.56 + 612.56 + 0.56 + 0.56 + 828.06 + 1,106.56 + 1,140.56 + 280.56 + 947.56 + 494.56 + 3,280.06 + 0.06 = 17,139.33
Step 6: Find the variance
- Divide the sum of squared deviations by the number of values in the data set (which is 16 in this case) to calculate the variance.
- Variance = 17,139.33 / 16 = 1,071.208125
Step 7: Find the standard deviation
- The standard deviation is the square root of the variance.
- Standard deviation = √(1,071.208125) ≈ 32.727
To find the range, standard deviation, and variance for the given sample data, follow these steps:
Step 1: Calculate the Range
- The range is the difference between the maximum and minimum values in the data set.
- Find the maximum value: 98
- Find the minimum value: 2
- Calculate the range as: maximum value - minimum value
Range = 98 - 2 = 96
Therefore, the range of the given sample data is 96.
Step 2: Calculate the Mean (Average)
- The mean is the sum of all values divided by the number of values.
- Add up all the values:
98 + 2 + 53 + 61 + 15 + 87 + 63 + 63 + 91 + 29 + 96 + 79 + 93 + 40 + 5 + 62 = 967
- Count the number of values: 16
- Calculate the mean as: sum of values / number of values
Mean (Average) = 967 / 16 = 60.44 (rounded to two decimal places)
Therefore, the mean (average) of the given sample data is 60.44.
Step 3: Find the Deviations from the Mean
- Subtract the mean from each value in the data set.
- Deviations from the mean:
98 - 60.44 = 37.56
2 - 60.44 = -58.44
53 - 60.44 = -7.44
61 - 60.44 = 0.56
15 - 60.44 = -45.44
87 - 60.44 = 26.56
63 - 60.44 = 2.56
63 - 60.44 = 2.56
91 - 60.44 = 30.56
29 - 60.44 = -31.44
96 - 60.44 = 35.56
79 - 60.44 = 18.56
93 - 60.44 = 32.56
40 - 60.44 = -20.44
5 - 60.44 = -55.44
62 - 60.44 = 1.56
Step 4: Calculate the Squared Deviations
- Square each deviation to eliminate negative values.
- Squared deviations:
(37.56)^2 = 1409.5236
(-58.44)^2 = 3417.3936
(-7.44)^2 = 55.2336
(0.56)^2 = 0.3136
(-45.44)^2 = 2066.4836
(26.56)^2 = 705.7536
(2.56)^2 = 6.5536
(2.56)^2 = 6.5536
(30.56)^2 = 933.7536
(-31.44)^2 = 986.0736
(35.56)^2 = 1263.3936
(18.56)^2 = 344.4736
(32.56)^2 = 1061.7536
(-20.44)^2 = 417.7936
(-55.44)^2 = 3072.7936
(1.56)^2 = 2.4336
Step 5: Calculate the Variance
- The variance is the average of the squared deviations from the mean.
- Add up all the squared deviations:
1409.5236 + 3417.3936 + 55.2336 + 0.3136 + 2066.4836 + 705.7536 + 6.5536
+ 6.5536 + 933.7536 + 986.0736 + 1263.3936 + 344.4736 + 1061.7536 + 417.7936
+ 3072.7936 + 2.4336 = 15210.4544
- Calculate the variance as: sum of squared deviations / number of values
Variance = 15210.4544 / 16 = 950.6534 (rounded to four decimal places)
Therefore, the variance of the given sample data is 950.6534.
Step 6: Calculate the Standard Deviation
- The standard deviation is the square root of the variance.
- Calculate the standard deviation as: square root of variance
Standard Deviation = √950.6534 = 30.83 (rounded to two decimal places)
Therefore, the standard deviation of the given sample data is approximately 30.83.