Section 2. 5: Measure of Position Quartiles Quartiles divide the data into 4 parts. Quartile Proportion of data behind this point Q1 1 Q2 1 Q3 3 4 2 4 Example 1: The test scores of 30 students are listed. Find the quartiles of the following data set. Steps: • Find Q2 (the median) • Split the data into 2 parts to the left and the right of the median. • Find the median of each part. These will be Q1 and Q3 Interquartile Range (IQR) The interquartile range is a measure of variation of the middle 50% of the data. Formula: IQR = Q3 − Q1 If any value of the data set exceeds 1.5 times the IQR, then that value is considered an outlier. Example 2: Find the IQR of the previous data set and find any outliers. The five-number summary consists of the quartiles of the data set, along with the max and min of the data set. We can represent the five-number summary visually by creating a Box-andWhiskers Plot. The next example goes over the construction of the plot. Example 3: Use the five-number summary; create a Box-and-Whisker plot from the data in example 1 Steps • Construct a number line that spans the range of the data • Plot the five numbers above the number line. • Draw a box above the number line from Q1 and Q3 . Draw a vertical line over Q2 . • Draw whiskers from the box to the minimum and maximum entries. Percentiles Percentiles divided the data set into 100 equal parts. Percentile is denoted as P95 . 95 in this example represent the 95th percentile • 95 percent of the data lies below this point • 5% percent of the data is above this point Example 4: Find P30 from the data set in example 1. Steps • Order the data if needed. • Find R which is given by the formula R =  • If R is an integer, then Pk =  k  n  100  xR + xR +1 . If R is not an integer, then round up R and Pk = xR 2 Standard Score The standard score, or z-score, represents the number of standard deviations a given value falls from the mean. Formula: z = Value - Mean x−µ = Standard Deviation σ Example 5: Find the z-score for the value 57, when the mean is 64 and the standard deviation is 5. Example 6: Find the z-score for the value 90, when the mean is 66 and the standard deviation is 7. Example 7: The test scores for a history test and a physics test are given below. Test History 79 4.5 Physics 69 3.7 Find the z-score for each test. On which test did the student perform better on?