In a neighborhood donut shop, One type of donut has 490 calories, Three types of donuts have 520 calories, Three types of donuts have 360 calories, five types of donuts have 350 calories, and four types of donuts have 390 calories. Find the standard deviation round your answer to the nearest 10th
To find the standard deviation, we need to follow these steps:
Step 1: Calculate the mean (average)
Step 2: Subtract the mean from each value and square the result
Step 3: Calculate the sum of squared differences
Step 4: Divide the sum by the number of values
Step 5: Take the square root of the result from step 4
Let's apply these steps to find the standard deviation for the calorie counts of the different types of donuts:
Step 1: Calculate the mean (average)
To calculate the mean, add up all the calorie counts and divide by the total number of donut types.
(490 + 3 * 520 + 3 * 360 + 5 * 350 + 4 * 390) / (1 + 3 + 3 + 5 + 4) = 456
So the mean is 456 calories.
Step 2: Subtract the mean from each value and square the result
For each type of donut, subtract the mean (456) from the calorie count and square the result. Here are the squared differences for each type:
(490 - 456)^2 = 14,440
(520 - 456)^2 = 4,096
(520 - 456)^2 = 4,096
(520 - 456)^2 = 4,096
(360 - 456)^2 = 9,216
(360 - 456)^2 = 9,216
(360 - 456)^2 = 9,216
(350 - 456)^2 = 11,376
(350 - 456)^2 = 11,376
(350 - 456)^2 = 11,376
(350 - 456)^2 = 11,376
(350 - 456)^2 = 11,376
(390 - 456)^2 = 4,356
(390 - 456)^2 = 4,356
(390 - 456)^2 = 4,356
(390 - 456)^2 = 4,356
Step 3: Calculate the sum of squared differences
Add up all the squared differences:
14,440 + 4,096 + 4,096 + 4,096 + 9,216 + 9,216 + 9,216 + 11,376 + 11,376 + 11,376 + 11,376 + 11,376 + 4,356 + 4,356 + 4,356 + 4,356 = 119,976
Step 4: Divide the sum by the number of values
Divide the sum of squared differences (119,976) by the total number of donut types:
119,976 / (1 + 3 + 3 + 5 + 4) = 5,998.8
Step 5: Take the square root of the result from step 4
Take the square root of 5,998.8 to find the standard deviation:
√5,998.8 ≈ 77.45
Rounding the standard deviation to the nearest 10th gives us approximately 77.5.
Therefore, the standard deviation of the donut calorie counts is 77.5.