five-number summary

measure of how the data are spread out. The data within the IQR comprise the middle 50% of your data.

## IQR = Q_{3 }- Q_{1 }

Let’s investigate the inter-quartile range for the combined scores for both Mr. Sneed’s and Mrs. Short’s students. If we were to make a single stemplot for the combined data, it would look like this:

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# Practice problem:

6. Identify the mean, median, mode and range for the combined data set above. [x bar= 75.6, median = 80, mode = 81, range = 98 – 32 = 66]

# Now compute Q_{1 }by finding the median for the data in the bottom half of

the data set. Do not include the median in the bottom half or the top half when calculating the middle of either the bottom half or the top half of the data set.

Here’s the bottom half of the data: 73, 77

32, 56, 61, 64, 65, 65, 67, 68, 71, 73,

# The median of this half is 66, so Q_{1 }= 66. What is Q_{3}? [Q3 = 83.5] Now

we know that 25% (one-fourth) of the students had scores ranging from 32 to 66; 50% of the students had scores between 66 and 83.5; and the top 25% of the students scored between 83.5 and 98. What would be another name for the median? [Q2] When we combine these five important dividing scores - the minimum score, the Q_{1 }score, the median, the Q_{3 }score

and the maximum score - we have an important summary of the data called the five-number summary:

# Five-number summary: min - Q_{1 }– median – Q_{3 }– max

For the data set above, the five-number summary is 32 – 66 – 80 – 83.5 – 98. A visual that helps to show how the scores are spread is found in a box- and-whisker plot, which is constructed from this five-number summary.

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