9. Measures of Center, Spread, and Position
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When is a value considered an outlier?
Outlier: 13
| Data set | Mean | Median | Mode | Range | Standard deviation |
|---|---|---|---|---|---|
| With Outlier | ≈ 35.8 | 36 | 36 | 38 | ≈ 9.3 |
| Without Outlier (13) | ≈ 37.3 | 36 | 36 | 29 | ≈ 7.5 |
Description: See solution.
If a value in a data set is more than 1.5 times the interquartile range away from the lower or upper quartiles, it is considered an outlier. Therefore, to identify any outliers, we first have to find these statistical measures, including any outliers.
Let's organize the information in a list using a graphing calculator. Press STAT and choose Edit.
Enter all of the data into the first list.
Next, analyze the data by pressing the STAT button again and navigating to the CALC menu. Press ENTER once to select the 1-Var Stats
option and then again to select the list in which we entered the data (usually L1). This will produce most of the statistical measures we are looking to find.
&40, 36, 29, 45, 51, 36, 48, 34 & 36, 22, 13, 42, 31, 44, 32, 34 We see that 36 occurs the most frequently, so this is the mode of the given data set.
To identify any outliers, we start by calculating the interquartile range (IQR). This is the difference between the upper and lower quartiles. Q_3- Q_1= IQR ⇔ 43- 31.5= 11.5 Next, we need to determine the maximum and minimum values for data to be considered an outlier. Outliers are more than 1.5 times the IQR away from the upper and lower quartiles. Q_1-1.5*IQR& ⇔ 31.5-1.5( 11.5)= 14.25 Q_3+1.5*IQR& ⇔ 43+1.5( 11.5)= 60.25 These calculations tell us that any values lower than 14.25 or greater than 60.25 are outliers. Since 13<14.25, 13 is an outlier.
Let's repeat the process, this time excluding 13 from the data set. We will type the new data set into another list, L2.
Once we have finished adding the new data set into L2, we can analyze the data the same way as before. After selecting the 1-Var Stats
option, however, we must remember to choose L2 by pushing 2nd and 2.
Examining the new output will give us most of the desired statistical measures. rcl x:& 37.33 &(Mean) σ x:& 7.45 &(Standard deviation) minX:& 22 &(Minimum value) Q_1:& 32 &(Lower quartile) med:& 36 &(Median) Q_3:& 44 &(Upper quartile) maxX:& 51 &(Maximum value) Range:& 51-22=29 & Note that 36 is still the most frequently occurring value. Therefore, the mode of the data set remains the same.
Finally, we summarize our findings in the table below so it is easier to compare the results.
| Data set | Mean | Median | Mode | Range | Standard deviation |
|---|---|---|---|---|---|
| With Outlier | ≈ 35.8 | 36 | 36 | 38 | ≈ 9.3 |
| Without Outlier (13) | ≈ 37.3 | 36 | 36 | 29 | ≈ 7.5 |
We can see that removing the outlier modified the mean, the range, and the standard deviation. Without the outlier the mean increased, which means that the automobiles are generally more efficient. Since the range and standard deviation decreased, the automobiles are more alike in fuel efficiency.