Seth asked a random sample of 20 students how much cash they had in their pocket. The following frequency table gives the responses.cash($) freq0 11 25 220 1344 1985 1Find the standard deviation. Give your answer to at least 2 decimal places
Question
Seth asked a random sample of 20 students how much cash they had in their pocket. The following frequency table gives the responses.cash($) freq0 11 25 220 1344 1985 1Find the standard deviation. Give your answer to at least 2 decimal places
Solution
To calculate the standard deviation, we first need to find the mean (average) of the data. Here are the steps:
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Multiply each cash value by its frequency to get the total cash for each category.
- 0*1 = 0
- 1*2 = 2
- 5*2 = 10
- 20*13 = 260
- 44*1 = 44
- 85*1 = 85
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Add these totals together to get the overall total cash: 0 + 2 + 10 + 260 + 44 + 85 = 401
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Divide this total by the total number of students (20) to get the mean: 401 / 20 = 20.05
Next, we calculate the variance, which is the average of the squared differences from the mean.
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Subtract the mean from each cash value, square the result, and multiply by the frequency for each category:
- (0 - 20.05)^2 * 1 = 401.0025
- (1 - 20.05)^2 * 2 = 724.402
- (5 - 20.05)^2 * 2 = 451.402
- (20 - 20.05)^2 * 13 = 0.0325
- (44 - 20.05)^2 * 1 = 574.8025
- (85 - 20.05)^2 * 1 = 4210.2025
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Add these totals together: 401.0025 + 724.402 + 451.402 + 0.0325 + 574.8025 + 4210.2025 = 6361.844
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Divide by the total number of students to get the variance: 6361.844 / 20 = 318.0922
Finally, we find the standard deviation, which is the square root of the variance.
- √318.0922 = 17.83
So, the standard deviation of the cash the students had in their pocket is approximately $17.83.
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