To find the number of standard deviations each value is from the mean, we first need to calculate the mean and standard deviation of the ticket sales.
Mean: (135 + 71 + 69 + 80 + 158 + 152 + 161 + 96 + 122 + 118 + 87 + 85) / 12 = 113
Next, we calculate the standard deviation. We will use the formula:
Standard Deviation = sqrt((Σ(xi - mean)^2) / N)
where:
- xi represents each individual value
- mean is the calculated mean value
- N is the total number of values
Calculate the squared differences from the mean for each value:
(135 - 113)^2 = 484
(71 - 113)^2 = 1764
(69 - 113)^2 = 1536
(80 - 113)^2 = 1089
(158 - 113)^2 = 2025
(152 - 113)^2 = 1521
(161 - 113)^2 = 2304
(96 - 113)^2 = 289
(122 - 113)^2 = 81
(118 - 113)^2 = 25
(87 - 113)^2 = 676
(85 - 113)^2 = 784
Sum these squared differences:
484 + 1764 + 1536 + 1089 + 2025 + 1521 + 2304 + 289 + 81 + 25 + 676 + 784 = 10598
Now, divide by N = 12:
10598 / 12 ≈ 883.17
Finally, take the square root to get the standard deviation:
sqrt(883.17) ≈ 29.72
Therefore, all the values fall within 2 standard deviations of the mean (113) since 113 +/- 2(29.72) = 113 +/- 59.44.
Susan keeps track of the number of tickets sold for each play presented at the community theater. Within how many standard deviations of the mean do all the values fall?
135, 71, 69, 80, 158, 152, 161, 96, 122, 118, 87, 85
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