Section 4.3 ~ Measures of Variation

Introduction to Probability and StatisticsMs. Young

Objective

Sec. 4.3

In this section you will be able to understand and interpret the following common measures of variation: Range The five-number summary (boxplot or box-and-whisker

plot) Standard deviation

Range Recall that the variation of a data set describes how

widely data are spread out about the center of the data set (low, moderate, or high)

The range of a data set is the difference between its highest and lowest data values

Example ~ the following data represents the wait time (in minutes) for 11 customers at two different banks

Big Bank: 4.1 5.2 5.6 6.2 6.7 7.2 7.7 7.7 8.5 9.3 11.0

Best Bank: 6.6 6.7 6.7 6.9 7.1 7.2 7.3 7.4 7.7 7.8 7.8

The range for Big Bank is 11.0 – 4.1 = 6.9 minutesThe range for Best Bank is 7.8 – 6.6 = 1.2 minutes

Since the range for Big Bank is much larger, this tells us that is has a higher variation (spread out wider)

Sec. 4.3

range = highest value (max) - lowest value (min)

Example 1 Computing the range can be useful at times, but can also be

misleading Example ~ Consider the following data sets which represent the quiz

scores for nine students. Which set has the greater range? Would you also say that this set has the greater variation?

Quiz 1: 1 10 10 10 10 10 10 10 10 Quiz 2: 2 3 4 5 6 7 8 9 10

The range for quiz 1 is 10 – 1 = 9 points The range for quiz 2 is 10 – 2 = 8 points

Quiz 1 has a higher range, but aside from the single outlier is has no

variation at all, therefore quiz 2 has the greater variation

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Quartiles Quartiles are values that divide the data distribution into

quarters

There are three quartiles, lower (Q1), middle (Q2), and upper (Q3) In order to find the quartiles, the values must be in ascending order

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Lower Quartile The lower quartile (Q1) is the median of the values in the lower

half of the data set If the data set is odd, exclude the middle value

Ex. ~

4.1 5.2 5.6 6.2 6.7 7.2 7.7 7.7 8.5 9.3 11.0

If the data set is even, split in half Ex. ~

Sec. 4.3

Q1

6.6 6.7 6.7 6.9 7.1 7.2 7.3 7.4 7.7 7.8

Q1

Middle Quartile The middle quartile (Q2) is the overall median of the data set

Odd data set: Ex. ~

4.1 5.2 5.6 6.2 6.7 7.2 7.7 7.7 8.5 9.3 11.0

Even data set: Ex. ~

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Q2

6.6 6.7 6.7 6.9 7.1 7.2 7.3 7.4 7.7 7.8

Q2

7.1 7.27.15

2

Upper Quartile The upper quartile (Q3) is the median of the values in the upper

half of the data set If the data set is odd, exclude the middle value

Ex. ~

4.1 5.2 5.6 6.2 6.7 7.2 7.7 7.7 8.5 9.3 11.0

If the data set is even, split in half Ex. ~

Sec. 4.3

Q3

6.6 6.7 6.7 6.9 7.1 7.2 7.3 7.4 7.7 7.8

Q3

Five-Number Summary

Once you know the quartiles, you can describe a distribution with a five-number summary, consisting of the low value, lower quartile, median (middle quartile), upper quartile, and high value Ex. ~ Write the five number summaries for the waiting times for Big

Bank and Best Bank.

Sec. 4.3

Boxplots The five-number summary can be displayed with a graph called a

boxplot (or box-and-whisker plot) The values from the lower to upper quartiles are enclosed in a box and

a line extends from the box to the low and high values which are considered the whiskers

Steps to Drawing a Boxplot Step 1: Draw a number line that spans all the data values in the data

set (usually leaving room on either end of the high and low values) Step 2: Enclose the values from the lower to the upper quartile in a

box Step 3: Draw a vertical line through the box at the median Step 4: Add “whiskers” by extending to the low and high values

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Example 2 A bakery collected the following data about the number of

loaves of fresh bread sold on each of 10 business days. Write a five-number summary and then make a boxplot to represent this data. State any skewness.

43 39 17 38 50 42 34 8 39 43

Sec. 4.3

Percentiles Percentiles divide a data set into 100 segments

They are essentially a rank of each individual data value The nth percentile of a data set divides the bottom n% of

data values from the top (100-n)% Example ~ The 35th percentile of a data set is the value that

separates the bottom 35% of data values from the top 65% If exam results stated that your exam score is in the 35th percentile,

that means that you scored higher than 35% of the people that took the exam and lower than 65% of the people that took the exam

If a data value lies between two percentiles it is often said to lie in the lower of the two percentiles

Example ~ If you score higher than 84.7% of all people taking a college entrance examination, it is said that you scored in the 84th percentile

The percentile of a data value is found using the following formula:

Sec. 4.3

number of values less than this data valuepercentile of data value = 100

total number of values in data set

Percentiles Cont’d…

Example ~ What percentile is the lower value, Q1 , Q2 , Q3 , and the upper value in for Big Bank?

Lower value = 4.1; there are no values lower than 4.1, so it is in the 0th percentile (0/11 = 0)

Q1 = 5.6; there are two values lower than 5.6 in a set of eleven values, so it is in the 18th percentile (2/11 = .1818…)

Q2 = 7.2; there are five values lower than 7.2 in a set of eleven values, so it is in the 45th percentile (5/11 = .4545…)

Q3 = 8.5; there are eight values lower than 8.5 in a set of eleven values, so it is in the 72nd percentile (8/11 = .7272…)

Upper value = 11.0; there are ten values lower than 11.0 in a set of eleven values, so it is in the 90th percentile (10/11 = .9090…)

Sec. 4.3

number of values less than this data valuepercentile of data value = 100

total number of values in data set

Example 3 Refer to table 4.4 on p.168 to answer the

following questions. What is the percentile for the data value of 592.79 ng/ml

for smokers? Since this is the 47th value, there are 46 values falling below

which means it will fall in the 92nd percentile (46/50 = .92) What values mark the 36th percentile in the data set on

p.168? Because we know that there are a total of 50 values in the

set, we can set up an equation to solve for the value

This means that the 18th value in the data set marks the 36th percentile, therefore 20.16 ng/ml (smokers) and 0.33 ng/ml (nonsmokers)

Sec. 4.3

36 10050

x .36

50

x .36(50) x 18x

Standard Deviation

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While the range and a five-number summary are both methods of explaining variation, the standard deviation is a single number that is most commonly used to describe variation of a data set This is universally accepted as the best measure of variation for a

statistical distribution The standard deviation is found by averaging the deviation from

each data value to the mean To calculate the standard deviation for a data set, follow these

steps: 1. Compute the mean of the data set 2. Find the deviation of each data value from the mean

Deviation from mean = data value – mean 3. Find the squares of all the deviations 4. Add all the squares 5. Divide this sum by the total number of data values minus 1 6. Take the square root of the value found in step 5

In general, the formula for the standard deviation is:2sum of (deviations from the mean)standard deviation =

total number of data values - 1

Example 4

1 5 6 8 11 316.2

5 5

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Find the standard deviation of the following data set.1 5 6 8 11

Step 1: Find the mean

Step 2: Find the deviations from the mean

1 – 6.2 = -5.2 5 – 6.2 = -1.2 6 – 6.2 = -0.28 – 6.2 = 1.811 – 6.2 = 4.8

Step 3: Square the deviations(-5.2)²= 27.04(-1.2)²= 1.44 (-0.2)² = .04(1.8)²= 3.24(4.8)²= 23.04

Example 4 Cont’d…

Sec. 4.3

Step 4: Add the squares

27.04 + 1.44 + .04 + 3.24 + 23.04 = 54.8

Step 5: Divide the sum by the total number of data values – 1

54.8/(5 – 1) = 54.8/4 = 13.7

Step 6: Take the square root of the value in step 5

The standard deviation is 3.7. This means that most values occur within 3.7 units in either direction of the mean of 6.2.

13.7 3.7

The Range Rule of Thumb

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The range rule of thumb is an approximation of the standard deviation

Example ~ Estimate the standard deviation of the data set used in the last example.

1 5 6 8 11

This approximation is not the best estimate in comparison to the actual standard deviation of 3.7, but is a rough estimate. It would be a much better representation if the data values were closer in number.

If you know the standard deviation of a data set, you can use the range rule of thumb to estimate the high and low values of a data set:

rangestandard deviation

4

11-1 10standard deviation 2.5

4 4

low value mean - (2 standard deviation)

high value mean + (2 standard deviation)

Example 5 Use the range rule of thumb to estimate the standard

deviations for the waiting times at Big Bank and Best Bank. Compare the estimates to the actual values in example 4 in the book (p.172). Big Bank:

Best Bank:

The actual standard deviations in example 4 are 1.96 and .44 respectively, so the range rule of thumb in in the right ballpark, but slightly underestimates it

Sec. 4.3

11-4.1 6.9standard deviation 1.725

4 4

7.8-6.6 1.2standard deviation .3

4 4

Example 6

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Studies of the gas mileage of a BMW under varying driving conditions show that it gets a mean of 22 miles per gallon with a standard deviation of 3 miles per gallon. Estimate the minimum and maximum typical gas mileage amounts that you can expect under ordinary driving conditions.

The range of gas mileage for the car is roughly from 16 to 28 miles per gallon.

low value mean - (2 standard deviation) = 22 - (2 3) = 16

high value mean + (2 standard deviation) = 22 + (2 3) = 28

Standard Deviation Using Summation Notation

Algebraic Form Definition

Sum of all values

Mean of a sample

Mean of an entire population.

Weighted Mean

Sec. 4.3

xx

( )

;

x w

w

x each value w weight

Recall from section 4.1, the notations used to describe mean:

The standard deviation formula using summation notation is as follows:

2( )

1

x xs

n