The shoe sizes of a group of middle school girls are shown. If a shoe size of 7 is added to the data, how does the IQR change?
Answer (2)
Arrange the original data in ascending order and calculate the original IQR: I QR = 1.75 .
Add the shoe size of 7 to the data set.
Calculate the new IQR: I Q R n e w = 1.5 .
Compare the two IQRs to find the change: C han g e = I Q R n e w − I QR = − 0.25 . The IQR decreases by 0.25.
− 0.25
Explanation
Arrange the original data First, we need to organize the original data in ascending order to easily find the quartiles. The original data is: 5.5, 6, 7, 8.5, 6.5, 6.5, 8, 7.5, 8, 5. Arranging this in ascending order, we get: 5, 5.5, 6, 6.5, 6.5, 7, 7.5, 8, 8, 8.5.
Calculate the IQR of the original data Next, we calculate the Interquartile Range (IQR) for the original data. The IQR is the difference between the third quartile (Q3) and the first quartile (Q1). With 10 data points, Q1 is the 25th percentile and Q3 is the 75th percentile. Q1 is the average of the 2nd and 3rd data points: Q 1 = 2 5.5 + 6 = 6.125 . Q3 is the average of the 8th and 9th data points: Q 3 = 2 8 + 7.5 = 7.875 . Therefore, the IQR for the original data is: I QR = Q 3 − Q 1 = 7.875 − 6.125 = 1.75 .
Add the new data point Now, we add the shoe size of 7 to the data set. The new data set is: 5, 5.5, 6, 6.5, 6.5, 7, 7, 7.5, 8, 8, 8.5.
Calculate the IQR of the new data Next, we calculate the IQR for the new data set. With 11 data points, Q1 is the 25th percentile and Q3 is the 75th percentile. Q1 is the value at position 0.25 * (11 + 1) = 3, so Q1 = 6.25. Q3 is the value at position 0.75 * (11 + 1) = 9, so Q3 = 7.75. Therefore, the IQR for the new data is: I Q R n e w = Q 3 − Q 1 = 7.75 − 6.25 = 1.5 .
Compare the IQRs and determine the change Finally, we compare the two IQRs to determine the change. The original IQR was 1.75, and the new IQR is 1.5. The change in IQR is: C han g e = I Q R n e w − I QR = 1.5 − 1.75 = − 0.25 . This means the IQR decreased by 0.25.
Final Answer The IQR changes by -0.25.
Real-world application Understanding IQR is very useful in real life. For example, in weather forecasting, IQR can be used to understand the variability in temperature measurements over a period of time. If you have daily high temperatures for a month, calculating the IQR can tell you how spread out the middle 50% of the temperatures are, giving you a sense of the temperature range you can typically expect. This is useful for planning activities and preparing for different weather conditions.
Examples
Understanding IQR is very useful in real life. For example, in weather forecasting, IQR can be used to understand the variability in temperature measurements over a period of time. If you have daily high temperatures for a month, calculating the IQR can tell you how spread out the middle 50% of the temperatures are, giving you a sense of the temperature range you can typically expect. This is useful for planning activities and preparing for different weather conditions.
The original IQR of the shoe sizes was calculated to be 1.5. After adding a shoe size of 7, the new IQR decreased to 1.375, resulting in a change of -0.125. This indicates less variability in the shoe sizes after including the new data point.
;
