What is the standard deviation of the following data? If necessary, round your answer to two decimal places.
46, 53, 38, 67, 46, 57, 61, 42, 63, 55, 46
To find the standard deviation of a data set, you can follow these steps:
Step 1: Find the mean (average) of the data set.
To find the mean, sum up all the numbers in the data set, and then divide the sum by the total number of data points.
Mean = (46 + 53 + 38 + 67 + 46 + 57 + 61 + 42 + 63 + 55 + 46) / 11
Mean = 581 / 11
Mean = 52.82 (rounded to two decimal places)
Step 2: Find the difference between each data point and the mean.
To do this, subtract the mean from each data point.
Difference between each number and the mean:
(46 - 52.82), (53 - 52.82), (38 - 52.82), (67 - 52.82), (46 - 52.82), (57 - 52.82), (61 - 52.82), (42 - 52.82), (63 - 52.82), (55 - 52.82), (46 - 52.82)
Step 3: Square each difference.
Square each of the differences calculated in Step 2.
Squared differences:
(-6.82)^2, (0.18)^2, (-14.82)^2, (14.18)^2, (-6.82)^2, (4.18)^2, (8.18)^2, (-10.82)^2, (10.18)^2, (2.18)^2, (-6.82)^2
Step 4: Find the mean of the squared differences.
To find the mean of the squared differences, sum up all the squared differences calculated in Step 3, and then divide the sum by the total number of data points.
Mean of the squared differences = (38.92 + 0.0324 + 219.0724 + 201.5524 + 38.92 + 17.4084 + 67.0724 + 117.3124 + 103.6324 + 4.7524 + 38.92) / 11
Mean of the squared differences = 848.38 / 11
Mean of the squared differences = 77.1263 (rounded to two decimal places)
Step 5: Find the square root of the mean of the squared differences.
To find the standard deviation, take the square root of the mean of the squared differences calculated in Step 4.
Standard deviation = √77.1263
Standard deviation ≈ 8.79 (rounded to two decimal places)
Therefore, the standard deviation of the given data set is approximately 8.79.
To find the standard deviation of a set of data, you need to follow these steps:
1. Find the mean (average) of the data set by adding up all the numbers and dividing by the total count.
In this case, the sum of the data set is: 46 + 53 + 38 + 67 + 46 + 57 + 61 + 42 + 63 + 55 + 46 = 614.
Since there are 11 numbers, the mean is: 614 / 11 = 55.82 (rounded to two decimal places).
2. Calculate the difference between each data point and the mean.
For example, the difference between 46 and 55.82 is: 46 - 55.82 = -9.82. Do this for each data point.
3. Square each difference calculated in step 2 to remove negative signs and emphasize larger variances.
For example, the squared difference for -9.82 is: (-9.82)^2 = 96.4324. Do this for each difference.
4. Find the mean of the squared differences.
Add up all the squared differences and divide by the total count.
In this case, the sum of the squared differences is: 96.4324 + 1.32 + 303.24 + 129.96 + 96.4324 + 0 + 19.4324 + 189.2324 + 53.2024 + 10.8324 + 96.4324 = 1085.95.
Since there are 11 numbers, the mean squared difference is: 1085.95 / 11 = 98.7227 (rounded to four decimal places).
5. Finally, take the square root of the mean squared difference to find the standard deviation.
The square root of 98.7227 is approximately 9.94 (rounded to two decimal places).
Therefore, the standard deviation of the given data set is approximately 9.94.
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.