Exercise 10 Page P28

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.

Analyzing the Data with any Outliers

Let's organize the information in a list using a graphing calculator. To do so, we will press STAT, and choose Edit. Then, we will enter all of the data into the first list.

Next, we will analyze the data by pressing the STAT button again and navigating to the CALC menu. Then, we 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.

To determine the range, we need to find the difference between the minimum and maximum values of the data set. The standard deviation is given as σ x. rcl x:& 21.8 &(mean) σ x:& 4.8 &(standard deviation) minX:& 10 &(minimum value) Q_1:& 19.5 &(lower quartile) med:& 22 &(median) Q_3:& 25.5 &(upper quartile) maxX:& 29 &(maximum value) Range:& 29-10 & & =19 & To find the mode, we have to identify the value that occurs most frequently. 25, 23, 21,27,29,19,10, 21,20,18,26, 23 We can see that 21 and 23 occur more frequently than the other values, so there are two modes in this set.

Identifying Outliers

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 ⇔ 25.5- 19.5= 6 Next, we need to determine the maximum and minimum values for data to be considered outlier. Outliers are more than 1.5 times the IQR away from the upper and lower quartiles. Q_1-1.5*IQR& ⇔ 19.5-1.5( 6)= 10.5 Q_3+1.5*IQR& ⇔ 25.5+1.5( 6)= 34.5 These calculations tell us that any values lower than 10.5 or greater than 34.5 are outliers. Therefore, 10 is an outlier.

Analyzing the Data without the Outliers

Let's repeat the process, excluding 10 from the data set. To do so, 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:& 22.9 &(mean) σ x:& 3.3 &(standard deviation) minX:& 18 &(minimum value) Q_1:& 20 &(lower quartile) med:& 23 &(median) Q_3:& 26 &(upper quartile) maxX:& 29 &(maximum value) Range:& 29-18 & & =11 & Note that 21 and 23 are still the most frequently occurring values. Therefore, modes of the data set remain the same.

Summary

Finally, we summarize our findings in the table below so it's easier to compare the results.