Section 3.2

Measures of Variation

Range Standard Deviation

Variance

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The Range

• The range is the difference between the largest and smallest values of a distribution.

• Example: Find the range: 10, 13, 17, 17, 18

The range = largest minus smallest = 18 -10 = 8

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The Standard Deviation

1n)xx(

s2

n = sample size

mean of the sample

Standard deviation of a sample

The standard variation is a measure of the average variation of the data entries from the mean.

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To calculate standard deviation of a sample

• Calculate the mean of the sample.• Find the difference between each entry (x) and the

mean. These differences will add up to zero.• Square the deviations from the mean.• Sum the squares of the deviations from the mean.• Divide the sum by (n 1) to get the variance.• Take the square root of the variance to get the • standard deviation.

1n

)xx(s

2

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The VarianceThe variance is the square of the standard

deviation

Variance of a Sample2

2 ( )

1

x xs

n

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ExampleFind the standard deviation and variance

x302622

2)x-(x xx 4 0-4

16 016___32 78

Sum = 0

Mean = 26

The variance2

2 ( )

1

x xs

n

= 32 2 =16

The standard deviation

s = 416 6

x

4

5

5

7

4

2)x-(x xx

Σx =25

1

0

0

2

1

1

0

0

4

1

mean = 5

ExampleFind the mean, the

standard deviation and variance

2( ) 6x x 3.2 / 7

Example cont.

Mean = 5

5.146

Variance

22.15.1deviationdardtanS

2( ) 61.5 1.22

1 4

x xs

n

2 1.5s 3.2 / 8

Computation Formulas for Sample Variance and Standard Deviation:

1nn

xxs devaition standard Sample

1nn

xxs variance Sample

22

22

2

To find ( Σx ) 2 Sum the x values, then square.

To find Σx2 Square the x values, then add.

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Use the computing formulas to find s and s2

5.115

5625131

s2

x

4

5

5

7

4

x2

16

25

25

49

1625 131 22.15.1s

22

2

1

xx nsn

2

2

1

xx nsn

10

Population Mean

2

N number of data values in the population

x

Nwhere

population mean

N number of data values in the population

x

Nwhere

Population Standard Deviation

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Coefficient Of Variation

• The disadvantage of the standard deviation as a comparative measure of variation is that it depends on the units of measurement. This means that it is difficult to use the standard deviation to compare measurements from different populations.

• For this reason, statisticians have defined the coefficient of variation, which expresses the standard deviation as a percentage of the sample or population mean.

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Coefficient Of Variation:• The coefficient of variation is a measurement of

the relative variability (or consistency) of data.

• Notice that the numerator and denominator in the definition of CV have the same units, so CV itself has no units of measurement. This give us the advantage of being able to directly compare the variability of two different populations using the coefficient of variation.

100or100x

sCV

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CV is used to compare variability or consistency

A sample of newborn infants had a mean weight of 6.2 pounds with a standard deviation of 1 pound.

A sample of three-month-old children had a mean weight of 10.5 pounds with a standard deviation of 1.5 pound.

Which (newborns or 3-month-olds) are more variable in weight?

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To compare variability, compare Coefficient of Variation

• For newborns:

• For 3-month-olds:

CV = 16%

CV = 14%

Higher CV: morevariable

Lower CV: more consistent

• You may wish to compare two groups of data, to answer:– Which is more consistent?– Which is more variable?

Use Coefficient of Variation

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ExampleA local fishing store sells spinners (a type of fishing lure).The store has only 8 different types of spinners for sale.The prices (in dollars) are2.10 1.95 2.60 2.00 1.85 2.25 2.15 2.25Find the coefficient of variationSolutiona. Compute the mean and standard deviation of the

population μ = $2.14 and σ = $0.22

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Example cont.b. Compare the CV of prices and comment on the

meaning of the results.

The CV can be though of as a measure of the spread of the data relative to the average of the data. Since the fishing store is very small, it carries a small selection of spinners that are all priced similarly. The CV tells us that the standard deviation of the spinner prices is only 10.28% from the mean.

0.22100 100 .1028 100 10.28%

2.14CV x x x

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ExampleA large fishing store in Nebraska has a broad selection

of spinners. The prices of a random sample of 10 spinners are

1.69 1.49 3.09 1.79 1.39 2.89 1.49 1.39 1.491.99

a. Use the calculator to compute and s = $0.62b. Compute the CV for the spinner prices

x and s

$1.87x

0.62100 100 .3316 100 33.16%

1.87CV x x x

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Example cont.Compare the mean, standard deviation, and CV for the

spinner prices at the two fishing stores. Comment on the differences.

The CV for Nebraska store is three times more than the CV from the previous example.

First, because the fishing store in the previous example is small, and tends to have higher prices (larger μ).

Second, it has limited selection of spinners with a smaller variation of price.

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Shebyshev’s TheoremThe spread of dispersion of a set of data about the mean

will be small if the standard deviation is small, and it will be large if the standard deviation is large. If we are dealing with a symmetrical bell-shaped distribution, then we can make very definite statements about the proportion of the data that must lie within a certain number of standard deviations on either side of the mean.

However, the concept of data spread about the mean can be expressed quite generally for all data distributions (skewed, symmetric, or other shape) by using the remarkable theorem of Chebyshev.

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CHEBYSHEV'S THEOREM

For any set of data and for any number k, greater than one, the proportion of the data that lies within k standard deviations of the

mean is at least:

2k

11

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Results of Chebyshev’s theorem

• For k = 2: or at least 75%of the data fall in the interval from

• from to (between 2 St Deviations)

• For K = 3 at least 88.9% (between 3 St Deviations)•

• For K = 4 at least 93.8% (between 4 St Deviations)

2 2

2 2

1 1 11 1 1 0.75 75

2 4k

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Using Chebyshev’s Theorem

• A mathematics class completes an examination and it is found that the class mean is 77 and the standard deviation is 6.

• According to Chebyshev's Theorem, between what two values would at least 75% of the grades be?

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Mean = 77 Standard deviation = 6

At least 75% of the grades would be in the interval:

s2xtos2x

77 – 2(6) to 77 + 2(6)

77 – 12 to 77 + 12

65 to 89Assignment 5 3.2 / 24

Entering Data (Calc.) Data is stored in Lists on the calculator. Locate and press the

STAT button on the calculator. Choose EDIT. The calculator will display the first three of six lists (columns) for entering data. Simply type your data and press ENTER. Use your arrow keys to move between lists.

Data can also be entered from the home screen using set notation -- {15, 22, 32, 31, 52, 41, 11} → L1 (where → is the STO key)

• Data can be entered in a second list based upon the information in a previous list. In the example below, we will double all of our data values in L1 and store them in L2. If you arrow up ONTO L2, you can enter a formula for generating L2. The formula will appear at the bottom of the screen. Press ENTER and the new list is created.

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Clearing Data (Calc.) • To clear all data from a list: Press STAT. From the EDIT

menu, move the cursor up ONTO the name of the list (L1). Press CLEAR. Move the cursor down. NOTE: The list entries will not disappear until the cursor is moved down. (Avoid pressing DEL as it will delete the entire column. If this happens, you can reinstate the column by pressing STAT #5 SetUpEditor.)

• You may also clear a list by choosing option #4 under the EDIT menu, ClrList. ClrList will appear on the home screen waiting for you to enter which list to clear. Enter the name of a list by pressing the 2nd button and the yellow L1 (above the 1).

To clear an individual entry: Select the value and press DEL.

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Sorting Data (Calc.) • Sorting Data: (helpful when finding the mode)

Locate and press the STAT button. Choose option #2, SortA(. Specify the list you wish to sort by pressing the 2nd button and the yellow L1 list name. Press ENTER and the list will be put in ascending order (lowest to highest). SortD will put the list in descending order.

• One Variable Statistical Calculations:Press the STAT button. Choose CALC at the top. Select 1-Var Stats. Notice that you are now on the home screen. Specify the list you wish to use by choosing the 2nd button and the list name: Press ENTER and view the calculations. Use the down arrow to view all of the information.

•

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One Variable Statistical Calculations (Calc.)

= mean = the sum of the data = the sum of the squares of the data = the sample standard deviation = the population standard deviation = the sample size (# of pieces of data) = the smallest data entry = data at the first quartile = data at the median (second quartile) = data at the third quartile = the largest data entry

2

1

3

min

max

x

x

x

x

x

s

n

X

Q

med

Q

X

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Measures of Dispersion (Calc) Range, Standard Deviation, Variance, Mean Absolute Deviation

• Problem: For the data set {10, 12, 40, 35, 14, 24, 13, 21, 42, 30},

find the range, the standard deviation, the variance, and the mean absolute deviation to the nearest hundredth.

• A quick reminder before we begin the solution:In statistics, the population form is used when the data being analyzed includes the entire set of possible data. The sample form is used when the data is a random sample taken from the entire set of data. You should use population form unless you know that you are working with a random sample of the data.

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Measures of Dispersion cont. (Calc)• To find the range: • To find the range:

Enter the data, as is, into L1. You can enter the list on the home screen and "store" to L1, or you can go directly to L1 (2nd STAT, #1 Edit).

• Sort the list to quickly retrieve the highest and lowest values for the range. (2nd STAT, #2 SortA). You can choose ascending or descending. Read the high and low values from L1 for computing the range.Range = 42 - 10 = 32.

• OR: To find the range: Do not sort. Simply type on the home screen using the min and max functions found under MATH → NUM #6 min and #7 max. Range = 32 3.2 / 30

Measures of Dispersion cont. (Calc)

• To find standard deviation: • To find standard deviation: Since this question deals with the

complete set, we will be using "population" form, not sample form.

• Go to one-variable stats for "population" standard deviation. STAT → CALC #1 1-Var Stats

• • NOTE! The standard deviations found in the CATALOG, stdDev,

and also found by 2nd LIST → MATH #7 stdDev are both Sample standard deviations.

• Population Standard Deviation = 11.433.2 / 31

Measures of Dispersion cont. (Calc)To find variance: To find variance: The "population" variance is the

square of the population standard deviation. The symbol is under VARS - #5 Statistics NOTE! The variance found in the CATALOG and also found by 2nd List → MATH #8 variance are both Sample variances.

To find mean absolute deviation: To find mean absolute deviation: To calculate the mean absolute deviation you will have to enter the formula.

Mean Absolute Deviation = 10.12

1

1| |

n

ii

Population MAD x xn

1

1| |

n

ii

Population MAD x xn

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Measures of Dispersion cont. (Calc)NOTE! Be sure that you have run 1-Var Stats (under STAT - CALC #1)

first, so that the calculator will have computed . Otherwise, you will get an error from this formula. and n are found under VARS #5 Statistics. Sum and abs are quickly found in CATALOG. Sum is also under 2nd LIST - MATH #5 sum. abs is also under MATH - NUM #1abs.

OR: To find mean absolute deviation: A longer, but workable, solution can also be accomplished using the lists. As stated above, run 1-Var Stats so the calculator will compute . Now, go to L2 (STAT #1 EDIT) and move UP onto L2. Type, at the bottom of the window, the portion of the formula that finds the difference between each data entry and the mean, using absolute value to make these distances positive. Now, find the mean, , of L2 by using 1-Var Stats on L2, and read the answer of 10.12.

x

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Measures of Dispersion on Grouped DataProblem: Data Entry Frequency

100 8 150 15 200 21 250 14 300 5

For the data set shown in this table, find the range, the standard deviation, and the variance to the nearest hundredth.

Since this question deals with the complete set, we will be using "population" form, not sample form.

For central tendency on grouped data, see Mean, Mode, Median with Grouped Data.

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Measures of Dispersion on Grouped Data• Solution: • To find the range: No need for calculator work for the range. It is easily

observed from the table.Range = 300 - 100 = 200.To find standard deviation: Remember, we are looking for "population" form which will be found using 1-Var Stats.

• Enter the "Data Entry" into L1 and the "Frequency" into L2. Go to one-variable stats to find "population" standard deviation. STAT → CALC #1 1-Var StatsBe sure to use parameters L1, L2 to indicate both the values AND their frequencies.

• NOTE! The standard deviation found in the CATALOG, stdDev, and also found by 2nd LIST → MATH #7 stdDev are both Sample standard deviations. Population Standard Deviation = 56.42

• Population Standard Deviation = 56.42 3.2 / 35

Measures of Dispersion on Grouped DataTo find variance: The "population" variance is the square of the

population standard deviation. The symbol is under VARS - #5 Statistics NOTE! The variance found in the CATALOG and also found by 2nd List → MATH #8 variance are both Sample variances.

Population Variance = 3183.42

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