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Statistics 9-12 / Week 01 / Thursday
4/6
Week 01 · Shape, Center and Spread

Thursday

Comparing two runners
// Lap times in Nova's log
⏱ about 20 min

Thursday: Comparing Two Runners

"My laps against yours," Comet says, laying two sheets side by side. "Same morning, same short loop."

Wren reads his own times. "49, 52, 54, 55, 57, 57, 57, 59, 60, 62, 65. Median 57, same as your mean. What do you notice?"

"Your box is wider," Comet says. "Your IQR is 6 and mine is 2. Same center, three times the spread."

"Standard deviation says the same thing," Wren says. "Mine is 4.3, yours is 2. I speed up and slow down."

Nova shows two box plots on one number line. "Would you like a hint? Look at the warm-up jog times too."

"The first morning has a long tail to the right," Comet says. "A few slow jogs pull the mean above the median."

"Skewed right," Wren says. "For a skewed set, report the median and IQR. They do not get pulled by the tail."

Two box plots, one number line

To compare two groups, put both box plots on the same scale. Compare the centers first, then the spreads.

Two box plots on one scale: Comet's narrow box around 56 and Wren's wider box around 57.
RunnerMeanMedianStandard deviationIQR
Comet565622
Wren57574.36

Both runners center near 56 seconds. Wren's spread is larger by both measures, so his pace varies more from lap to lap.

Shape decides which measures to report

A symmetric set has a hill that is even on both sides. The mean and median agree, so report the mean and standard deviation.

A skewed set has a long tail on one side. The tail pulls the mean toward it, but the median stays put.

Skewed right: the tail points to larger values and the mean is above the median. Skewed left: the tail points to smaller values.

For a skewed set, report the median and IQR. They are built from positions, so a long tail cannot drag them.

A dot plot of warm-up jog times from 40 to 52 seconds, bunched low with a long tail to the right.
A dot plot of warm-up jog times from 40 to 52 seconds, bunched high with a tail to the left.
MorningMeanMedianShape
First4443skewed right
Second47.649skewed left
COMPARE THE GROUPS
  • Read the question.
  • Tap your answer.
Comet's laps: 52, 54, 55, 55, 56, 56, 56, 57, 57, 58, 60. Wren's laps: 49, 52, 54, 55, 57, 57, 57, 59, 60, 62, 65. Who has the larger interquartile range?
Comet's laps: 52, 54, 55, 55, 56, 56, 56, 57, 57, 58, 60. Wren's laps: 49, 52, 54, 55, 57, 57, 57, 59, 60, 62, 65. Who has the larger standard deviation?
The first morning's jog times are 40, 41, 41, 42, 42, 43, 43, 44, 46, 50, 52 seconds. What shape does the distribution have?
The second morning's jog times are 40, 42, 46, 47, 48, 49, 49, 50, 50, 51, 52 seconds. What shape does the distribution have?
WHICH MEASURES TO REPORT?
  • Read the question.
  • Tap your answer.
The first morning's jog times are skewed right. Which center and spread describe them best?
Comet's laps are symmetric with no extreme value. Which center and spread describe them best?
A set is skewed right. Which is larger, the mean or the median?

How the standard deviation is built

  1. Find the mean of the values.
  2. Subtract the mean from each value to get its distance from the mean.
  3. Square each distance so that no distance is negative.
  4. Add the squared distances and divide by n, the number of values.
  5. Take the square root to return to the original units.
THE STANDARD DEVIATION, IN ORDER
  • ?Find the mean
  • ?Subtract the mean from each value
  • ?Square each distance
  • ?Add the squares and divide by n
  • ?Take the square root
WHY THIS EXERCISEEvery step has a reason. Squaring removes the minus signs, and the root brings back seconds.
A set with a long tail to the right is called skewed what? Type one word.
Which center stays put when a tail pulls the data: the mean or the median? Type one word.

Careful comparing. Tomorrow you find shape, center and spread in everyday counts around the loop.

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