GrowingKnowing.com © 2011 1. Percentile What ‘s the difference between percentile and percent? Percent measures ratio 90% percent on a test shows you

GrowingKnowing.com © 2011

1GrowingKnowing.com © 2011

PercentileWhat ‘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 © 2011 2

FormulaThere 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 belowwhich is different from Excel’s formula.

Formula i = percentile/100 x nSort 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


Example 1Find the 50th percentile for : 1, 4, 2, 5, and 4

Sort the data 1st: 1, 2, 4, 4, 5n = 5 because we have 5 data itemsCalculate 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.


Example 2: percentile. Find 25th percentile for numbers: 3, 9, -1,

and 0 Sort the data: -1, 0, 3, 9i = 25/100 x 4 = 1i is whole number, so take average of data

item in position 1 and the next data item in position 2.(-1 + 0) / 2 = -0.525th percentile = -0.5


Interquartile Range (IQR)TIP: Review percentiles before you study IQRIQR range shows you the middle 50 percentile of

your data from the 25th percentile to 75th 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.


25th 50th 75th

IQR

Calculate IQRThere 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 25th percentile = 4Calculate Q3: the 75th percentile = 8IQR = 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.


OutliersIQR 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 dataBook have been written about outliers as the

exceptional performersWho 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 ?


Outlier calculationIf first quartile is 10, third quartile is 30, what

are the outliers?IQR = 30 – 10 = 20Upper outlier = Q3+ 1.5(IQR) = 30 + 1.5(20) > 60Lower outlier = Q1 – 1.5(IQR) = 10 - 1.5(20) < -20Any 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 © 2011 9

Box PlotA 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.


9322

675747

Q3

Q2

Q1

Symmetrical data

Box PlotIf 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


Skewed left Skewed right

z scoreZ score measures position by the number of standard

deviations (S.D.) above or below the mean.Negative z score is S.D. below the meanPositive z score is S.D. above the mean

Formulaz = (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.


z scoreExcel: =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


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GrowingKnowing.com © 2011 1. Percentile What ‘s the difference between percentile and percent? Percent measures ratio 90% percent on a test shows you