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A meteorologist recorded the following temperatures 71 74 81 78 67 69 64 today's temperature is 67 if today's temperature is included in the data set how does it change the interquartile range

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To determine how today's temperature of 67 affects the interquartile range (IQR) of the data set, we first need to calculate the IQR of the original data set and then include the new temperature to find the updated IQR.

Original Data Set

The original temperatures recorded are: 71, 74, 81, 78, 67, 69, 64

  1. Sort the Data: 64, 67, 69, 71, 74, 78, 81

  2. Find the Quartiles:

    • The median (Q2) is the middle value. Since there are 7 values, the median is the 4th number: \[ Q2 = 71 \]
    • The lower quartile (Q1) is the median of the first half of the data (64, 67, 69): \[ Q1 = 67 \]
    • The upper quartile (Q3) is the median of the second half of the data (74, 78, 81): \[ Q3 = 78 \]

  3. Calculate the Interquartile Range (IQR): \[ IQR = Q3 - Q1 = 78 - 67 = 11 \]

Updated Data Set

Now, let's include today's temperature of 67: 64, 67, 67, 69, 71, 74, 78, 81

  1. Sort the Updated Data: 64, 67, 67, 69, 71, 74, 78, 81

  2. Find the Quartiles:

    • The median (Q2) is now the average of the 4th and 5th values (69 and 71): \[ Q2 = \frac{69 + 71}{2} = 70 \]
    • The lower quartile (Q1) is the median of the first half (64, 67, 67, 69): \[ Q1 = \frac{67 + 67}{2} = 67 \]
    • The upper quartile (Q3) is the median of the second half (71, 74, 78, 81): \[ Q3 = \frac{74 + 78}{2} = 76 \]

  3. Calculate the New Interquartile Range (IQR): \[ IQR = Q3 - Q1 = 76 - 67 = 9 \]

Conclusion

  • The original IQR was 11.
  • The new IQR after including today's temperature of 67 is 9.

Change in IQR: The IQR decreases from 11 to 9.