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?
137, 134, 91, 61, 150, 155, 110, 148, 90, 169, 67, 61
* 5
* 4
* 2
* 3
The answer is 2 i just took the quick check
To determine within how many standard deviations of the mean all the values fall, we need to calculate the standard deviation of the given set of numbers and then determine which values fall within that range.
Here's how you can calculate it:
Step 1: Calculate the Mean (Average)
- Add up all the values: 137 + 134 + 91 + 61 + 150 + 155 + 110 + 148 + 90 + 169 + 67 + 61 = 1373
- Divide the sum by the number of values (12): 1373 / 12 = 114.42 (rounded to two decimal places)
Step 2: Calculate the Difference from the Mean for each number
- Subtract the mean from each value and square the result:
(137 - 114.42)^2 = 506.13
(134 - 114.42)^2 = 382.60
(91 - 114.42)^2 = 547.16
(61 - 114.42)^2 = 2868.49
(150 - 114.42)^2 = 1283.88
(155 - 114.42)^2 = 1664.67
(110 - 114.42)^2 = 19.06
(148 - 114.42)^2 = 1136.45
(90 - 114.42)^2 = 594.34
(169 - 114.42)^2 = 3014.95
(67 - 114.42)^2 = 2225.55
(61 - 114.42)^2 = 2868.49
Step 3: Calculate the Variance
- Add up all the squared differences calculated in step 2 and divide by the number of values minus 1:
(506.13 + 382.60 + 547.16 + 2868.49 + 1283.88 + 1664.67 + 19.06 + 1136.45 + 594.34 + 3014.95 + 2225.55 + 2868.49) / (12 - 1) = 7305.099
Step 4: Calculate the Standard Deviation
- Take the square root of the variance calculated in step 3:
√7305.099 ≈ 85.51 (rounded to two decimal places)
Now that we have the standard deviation, we can determine within how many standard deviations of the mean all the values fall. To do that, we consider the empirical rule or the 68-95-99.7 rule:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
Since the question is asking within how many standard deviations all the values fall, we can conclude that the answer is 2, as approximately 95% of the values will fall within two standard deviations of the mean.
First calculate the mean. It is 114.4.
Then calculate the standard deviation. Call it sigma. I'll leave that up to you.
The number with the biggest deviation from the mean is 169, which is 54.6 away from the mean.
It looks to me like all numbers are within 2 sigma of the mean. But this means they are also within 3, 4 or 5 sigma. This is a poorly worded question in my opinion.
to find the mean, add up all your numbers and divide the sum by 12
To find the standard deviation proceed as follows:
1. take the difference between the mean and each data value
2. Square that difference, which means it didn't make any difference if the difference was + or -
3. Add up all those squared differences.
4. Divide that sum by N, or by (N-1), where N is the number of data values, in your case N=12
You will have to check with your text, your instructor or your course outline to see which method you use.
This result is called the variance,
5. take the √ of that result.
see ...
http://davidmlane.com/hyperstat/A16252.html