Position Growing Knowing com 2011 1 Percentile What

Position Growing. Knowing. com © 2011 1

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. Growing. Knowing. com © 2011 2

Formula �There are 3 popular formulas for percentiles � Check with your instructor to ensure you use the right formula. � In Excel, =Percentile(A 1: A 9, percentile) � where your data is in cells A 1 to A 9, 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 Growing. Knowing. com © 2011 3

Example 1 �Find the 50 th 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 50 th percentile or the middle position, and as you can see the formula we are using confirms our Median calculations. Growing. Knowing. com © 2011 4

Example 2: percentile. �Find 25 th 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 Growing. Knowing. com © 2011 5

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. 25 th 50 th 75 th IQR �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. Growing. Knowing. com © 2011 6

Calculate IQR �There is no Excel function for IQR �Use Excel like a fancy calculator to determine IQR. �Example: What is the IQR if Q 1 is 75 and Q 3 is 200 �Answer: IQR = Q 3 – Q 1 = 200 – 75 = 125 �What is the IQR for this data: 6, 2, 3, 4, 9, 10, 5, 7, 8? �Calculate Q 1: the 25 th percentile = 4 �Calculate Q 3: the 75 th percentile = 8 �IQR = Q 3 – Q 1 = 8 – 4 = 4 �TIP: There a few ways to calculate percentiles so ensure you are using the method recommended by your teacher. Growing. Knowing. com © 2011 7

Outliers �IQR is very important in the calculation of outliers. �An outlier calculation is at least 1. 5 times the IQR � Lower outlier = Q 1 – 1. 5(IQR) � Upper outlier = Q 3 + 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 ? Growing. Knowing. com © 2011 8

Outlier calculation �If first quartile is 10, third quartile is 30, what are the outliers? �IQR = 30 – 10 = 20 �Upper outlier = Q 3+ 1. 5(IQR) = 30 + 1. 5(20) > 60 �Lower outlier = Q 1 – 1. 5(IQR) = 10 - 1. 5(20) < -20 �Any data value more than 60 or less than -2 o 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. Growing. Knowing. com © 2011 9

Box Plot �A box plot shows the quartiles using a visual picture of a box. � The left side of the box is Q 1, � a line in the middle of the box is Q 2 (i. e. median), � and Q 3 is the right side of the box �Lines, called whiskers, that are drawn from the box to the minimum and maximum values. Q 1 Q 2 Q 3 22 Symmetrical data 93 47 57 Growing. Knowing. com © 2011 67 10

Box Plot �If the whiskers are longer on one side than another, that indicates the distribution skewed to the long side. �If the median (Q 2) is to the left of the middle, the distribution is skewed right. �If Q 2 is to the right of the middle, the distribution is skewed left Skewed left Growing. Knowing. com © 2011 Skewed right 11

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. Growing. Knowing. com © 2011 12

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 Growing. Knowing. com © 2011 13