"Where is the middle of my laps?" Comet asks. "And how spread out are they? I want two numbers."
"Two ways to answer," Wren says. "Way one: the mean and the standard deviation. Way two: the median and the IQR."
He adds the eleven times. "616 seconds in all. Divide by 11: the mean is 56."
"And the median was 56," Comet says. "Same number. What do you notice?"
"The hill is even on both sides," Wren says. "When a set is symmetric, the two centers agree."
Nova projects each lap's distance from the mean. "Would you like a hint? Square the distances before you average them."
"Average of the squares, then the square root," Wren says. "The crew divides by n. Standard deviation 2 seconds."
"And the IQR is 57 minus 55, which is 2," Comet says. "The middle half of my laps sits inside 2 seconds."
The mean is the sum divided by the count: 616 ÷ 11 = 56 seconds.
The standard deviation measures a typical distance from the mean. Find each value's distance from the mean and square it.
Average the squared distances. The crew's rule is to divide by n, the number of values. Then take the square root.
| Lap time | Distance from mean | Squared distance |
|---|---|---|
| 52 | -4 | 16 |
| 54 | -2 | 4 |
| 55 | -1 | 1 |
| 55 | -1 | 1 |
| 56 | 0 | 0 |
| 56 | 0 | 0 |
| 56 | 0 | 0 |
| 57 | 1 | 1 |
| 57 | 1 | 1 |
| 58 | 2 | 4 |
| 60 | 4 | 16 |
The squared distances add to 44. Divide by 11: 4. The square root is 2.
So a typical lap sits about 2 seconds from the mean of 56.
The median is the middle value in order: 56. The quartiles are the medians of the lower and upper halves: 55 and 57.
The interquartile range is 57 - 55 = 2 seconds. The middle half of Comet's laps spans 2 seconds.
The median and IQR use only positions in the ordered list. The mean and standard deviation use every value's size.
| Measure | Center | Spread | Uses |
|---|---|---|---|
| Way 1 | mean 56 | standard deviation 2 | the size of every value |
| Way 2 | median 56 | IQR 2 | positions in the ordered list |
| Statement | True or false? |
|---|---|
| 616 ÷ 11 = 56 | ? |
| The crew finds the standard deviation by dividing the squared distances by n. | ? |
| The median uses the size of every value in the set. | ? |
| 57 - 55 = 2 | ? |
Two ways that agree today. Tomorrow is Data Lab: your own laps, your own plots.