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2. Percentile What ‘s the difference between percentile and percent? Percent measures ratio 90% percent on a test shows you got 9 out 10 questions right. Percentile measures position 90 percentile shows you beat 9 out of 10 students on the test, even if the whole class failed that test. Business uses percentiles to reward the best employees, best customers, or possibly drop the worst. GrowingKnowing.com © 20112

3. Formula There are 3 popular formulas for percentiles Check with your instructor to ensure you use the right formula. In Excel, =Percentile(A1:A9, percentile) where your data is in cells A1 to A9, and percentile is written as a decimal (i.e. 90 percentile is written as.9) Our practice questions uses the more popular formula below which is different from Excel’s formula. Formula i = percentile/100 x n Sort data If i is a decimal, round up to the next data item in the sorted list If i is a whole number, take the average of data item i plus the next data item in the sorted list. n is the number of data items in your list GrowingKnowing.com © 20113

4. Example 1 Find the 50th percentile for : 1, 4, 2, 5, and 4 Sort the data 1 st : 1, 2, 4, 4, 5 n = 5 because we have 5 data items Calculate i = 50/100 x 5 = 2.5 Since i is a decimal, we round up to the next data item which is position 3 in the sorted list. The number 4 is in position 3 of our sorted list. Note: Median is another name for 50th percentile or the middle position, and as you can see the formula we are using confirms our Median calculations. GrowingKnowing.com © 20114

5. Example 2: percentile. Find 25th percentile for numbers: 3, 9, -1, and 0 Sort the data: -1, 0, 3, 9 i = 25/100 x 4 = 1 i is whole number, so take average of data item in position 1 and the next data item in position 2. (-1 + 0) / 2 = -0.5 25 th percentile = -0.5 GrowingKnowing.com © 20115

6. Interquartile Range (IQR) TIP: Review percentiles before you study IQR IQR range shows you the middle 50 percentile of your data from the 25 th percentile to 75 th percentile. We often want to know the middle 50 percentile of the data: we design a chair or other new product to fit the middle segment of people and ignore the extremes. GrowingKnowing.com © 20116 25 th 50 th 75th IQR

7. Calculate IQR There is no Excel function for IQR Use Excel like a fancy calculator to determine IQR. Example: What is the IQR if Q1 is 75 and Q3 is 200 Answer: IQR = Q3 – Q1 = 200 – 75 = 125 What is the IQR for this data: 6, 2, 3, 4, 9, 10, 5, 7, 8? Calculate Q1: the 25 th percentile = 4 Calculate Q3: the 75 th percentile = 8 IQR = Q3 – Q1 = 8 – 4 = 4 TIP: There are a few ways to calculate percentiles so ensure you are using the method recommended by your teacher. GrowingKnowing.com © 20117

8. Outliers IQR is very important in the calculation of outliers. An outlier calculation is at least 1.5 times the IQR Lower outlier = Q1 – 1.5(IQR) Upper outlier = Q3 + 1.5(IQR) Outliers indicate the exceptional data Book have been written about outliers as the exceptional performers Who are our outstanding sales people? What bacteria is exceptionally lethal? Performance can be exceptionally good or bad! The outlier calculation helps you determine what exactly is the point that signals outstanding performance. Is 10 good enough, or do you need to score 12.3 ? GrowingKnowing.com © 20118

9. Outlier calculation If first quartile is 10, third quartile is 30, what are the outliers? IQR = 30 – 10 = 20 Upper outlier = Q3+ 1.5(IQR) = 30 + 1.5(20) > 60 Lower outlier = Q1 – 1.5(IQR) = 10 - 1.5(20) < -20 Any data value more than 60 or less than -2o is an outlier (which is an extreme value) Some books use 3(IQR) as a more extreme outlier. You can now calculate if your friend is an extreme video gamer or just plays a lot. GrowingKnowing.com © 20119

10. Box Plot A box plot shows the quartiles using a visual picture of a box. The left side of the box is Q1, a line in the middle of the box is Q2 (i.e. median), and Q3 is the right side of the box Lines, called whiskers, that are drawn from the box to the minimum and maximum values. GrowingKnowing.com © 201110 93 22 67 57 47 Q3 Q2 Q1 Symmetrical data

11. Box Plot If the whiskers are longer on one side than another, that indicates the distribution skewed to the long side. If the median (Q2) is to the left of the middle, the distribution is skewed right. If Q2 is to the right of the middle, the distribution is skewed left GrowingKnowing.com © 201111 Skewed left Skewed right

12. z score Z score measures position by the number of standard deviations (S.D.) above or below the mean. Negative z score is S.D. below the mean Positive z score is S.D. above the mean Formula z = (x – mean) / standard deviation where x is any data value you select or want Brackets are needed or calculators/computers will divide first then subtract which gives an incorrect answer. GrowingKnowing.com © 201112

13. z score Excel: =STANDARDIZE(x, mean, standard deviation) Since a z score is a subtraction and a division, it may be faster to calculate z mentally than use a computer. What is the z score for 50 if you have a mean of 100 and standard deviation of 10? z score = (50 - 100) / 10: = -5 What is the z score for 150 if the mean is 100 and standard deviation is 25? z score = (150 - 100) / 25: = +2 GrowingKnowing.com © 201113