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What are the upper and lower quartiles of this data set?

118, 124, 124, 137, 138, 138, 138, 140, 143, 150, 151, 152, 156, 158, 159, 159, 159, 160, 162, 163, 165, 169, 170, 172, 173, 174, 177, 178, 206

Answer :

Answer:

Lower quartile = 139

Upper quartile = 169.5

Step-by-step explanation:

Step 1: Find the median:

  • Before we can find the upper and lower quartiles, we need to find the median, since it'll help us when finding the upper and lower quartiles.
  • The median lies in the middle of a data set and thus it has the same number of data points to the left and right.

Since there are 29 data points in this data set, there will be 14 points to the left and right of the median:

118, 124, 124, 137, 138, 138, 138, 140, 143, 150, 151, 152, 156, 158, 159, 159, 159, 160, 162, 163, 165, 169, 170, 172, 173, 174, 177, 178, 206

Since (the first) 159 has 14 points to the left and right of it, it's the median.

Step 2: Find the lower quartile:

  • For the lower quartile, 25% of the data lies below it and 75% of the data lies above it.

To find the lower quartile for this data set, we find the middle term between the point(s) below the median:

118, 124, 124, 137, 138, 138, 138, 140, 143, 150, 151, 152, 156, 158

  • Since there are 14 data points below the median, we get two numbers in the middle, as 6 data points lie to the left of 138 and 6 six data points lie to the right of 140.

Thus, we find the lower quartile by averaging these two numbers:
Lower quartile = (138 + 140) / 2

Lower quartile = 278 / 2

Lower quartile = 139

Thus, the lower quartile of the data set is 139.

Step 3: Find the upper quartile:

  • For the upper quartile, 75% of the data lies below it and 25% of the data lies above it.

To find the upper quartile, we find the middle term between the point(s) above the median:

159, 159, 160, 162, 163, 165, 169, 170, 172, 173, 174, 177, 178, 206

  • Since there are 14 data points below the median, we get two numbers in the middle, as 6 data points lie to the left of 169 and 6 six data points lie to the right of 170.

Thus, we find the upper quartile by averaging these two numbers:

Upper quartile = (169 + 170) / 2

Upper quartile = 339/2

Upper quartile = 169.6

Thus, the lower quartile of the data set is 169.6.

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