A Look at Means, Variances, Standard Deviations, and z-Scores Dr. Margie Mason College of William...

A Look at Means, Variances, Standard Deviations, and z-Scores

Dr. Margie MasonCollege of William and Mary

[email protected]

Based on theTechnical Assistance Document

2009 Algebra I Standard of Learning A.9

http://www.doe.virginia.gov/instruction/high_school/mathematics/technical_assistance_algebra1_a9.pdf

Algebra Standard of Learning A.9

The student, given a set of data, will interpret variation in real-world

contexts and calculate and interpret mean absolute deviation, standard

deviation, and z-scores.

Mathematics SOL 5.165.16 The student willa) describe mean, median, and mode as measures of center;b) describe mean as fair share;c) find the mean, median, mode, and range of a set of data; andd) describe the range of a set of data as a measure of variation.

Algebra SOL A.10The student will compare and contrast multiple univariate data sets, using box-and-whisker plots.

Algebra SOL A.10Let’s use unifix stick heights of

5 6 8 8 1013 15 17 18 20

Determining a Box-and-Whisker Plot

1. Find:Median – the middle value when arranged from smallest to largest. 11.5Lower extreme (LE) – the smallest value 5

Upper extreme (UE) – the largest value 20Lower quartile (LQ) – the value halfway between the lower extreme and the median. 8Upper quartile (UQ) – the value halfway between

the upper extreme and the median. 17Interquartile range (IQR) – The difference between the upper quartile and the lower quartile. 9


2. Determine and draw the scale.Subtract the smallest value from the largest value to determine the range. Choose a reasonable size for the intervals based on the range to be covered, e.g., 1, 2, 5, or 10. Draw the scale much like a number line at the bottom of your plot.


3. Draw the box:a. Length of the box extends for LQ to

UQ. It is drawn above the scale.b. Mark the median.c. Width of box can be anything.


4. Draw the whiskers:a. Draw from the box you just

drew to LE and UE.


Alternate method for drawing the whiskers:4. Determine the outliers: Multiply the IQR by 1.5 and add this number to the UQ and subtract it from the LQ. Any values outside these limits are outliers.


Alternate Method:5. Draw whiskers:

a. Draw from box to LE and UE excluding

outliers.b. Place asterisks on any outliers.


How many years of experience as a teacher do you have?

Mathematics SOL 6.15The student will a) describe mean as balance point; andb) decide which measure of center is appropriate for a given purpose.

Mathematics SOL 6.15The mean is the numerical average of the data set and is found by adding the numbers in the data set together and dividing the sum by the number of data pieces in the set.In grade 5 mathematics, mean is defined as fair-share.

Mathematics SOL 6.15Mean can be defined as the point on a number line where the data distribution is balanced. This means that the sum of the distances from the mean of all the points above the mean is equal to the sum of the distances of all the data points below the mean. This is the concept of mean as the balance point.Defining mean as balance point is a prerequisite for understanding standard deviation.

Mathematics SOL 6.15Balance Point: The sum of the distances on a number line from the mean of all the points above the mean is equal to the sum of the distances of all the data points below the mean.

7 + 6 + 4 + 4 + 2 = 1 + 3 + 5 + 6 + 8

Sample vs. Population DataA statistical population includes all elements in the set. A sample is a subset of the population.A data set, whether a sample or population, is comprised of individual data points referred to as elements of the data set.Start with small defined population data sets of approximately 30 items or less.

*Elements*An element of a data set will be represented as xi. Where i represents the ith term of the data set.

Sample vs. Population DataThe arithmetic mean of a population is represented by the Greek letter (mu), while the calculated arithmeticmean of a sample is represented by , read “x bar.”

Mean Absolute Deviation vs. Variance and Standard Deviation

Measuring dispersion or spread of a data set about the mean

One measure of spread is to find the sum of the deviations between each element and the mean; however, this sum is always zero.


Two methods:take the absolute value of the deviations before finding the average, (Mean Absolute Deviation) orsquare the deviations before find the average (Variance and Standard Deviation)


Summation Notation

Mean Absolute DeviationThe arithmetic mean of the absolute values of the deviations of elements from the mean of a data set.

5 6 8 8 10 13 15 17 18 20

|5-12| + |6-12| + |8-12| + |8-12| +|10-12| + |13-12| + |15-12| +|17-12| + |18-12| + |20-12| 10

= 7 + 6 + 4 + 4 + 2 + 1 + 3 + 5 + 6 + 810

= 46 10= 4.6

*Mean Absolute Deviation*Mean absolute deviation =

where represents the mean of the data set, n represents the number of elements in the data set, and xi represents the ith element of the data set.

Mean Absolute DeviationMean absolute deviation is less affected by outlier data than the variance and standard deviation. Outliers are elements that fall at least 1.5 times the interquartile range (IQR) below the first quartile (Q1) or above the third quartile (Q3).

VarianceThe average of the squared deviations from the mean is known as the variance and is another measure of the spread of the elements in the set.

(5-12)2 + (6-12)2 + (8-12)2 + (8-12)2 +(10-12)2 + (13-12)2 + (15-12)2 +(17-12)2 + (18-12)2 + (20-12)2

10

= (-7)2 + (-6) 2 + (-4) 2 + (-4) 2 + (-2) 2 + 12 + 32 + 52 + 62 + 82

10 = 49 + 36 + 16 + 16 + 4 + 1 + 9 + 25 + 36 + 64 = 256 = 25.6

10 10

*Variance*

Variance ( s 2)= where

represents the mean of the data set, n represents number of elements in the data set, and

xi represents the ith element of the data set.

VarianceThe differences are squares so that they don’t cancel each other out when finding the sum. When squaring the differences, the units of measure are squared and the larger differences are “weighted” more heavily than smaller differences. In order to provide a measure of variation in terms of the original units of the date, the square root of the variance is taken, yielding the standard deviation.

Standard Deviation

The positive square root of the variance of the data set. The greater the value, the more spread out the data are about the mean. The lesser (closer to 0) the value, the closer the data are clustered about the mean.

*Standard Deviation*

Standard deviation (s)= where

represents the mean of the data set, n represents the number of elements in the data set, and xi represents the ith element of the data set.

Standard Deviation

“ ”, written and read “sigma”, represents the standard deviation of a population and “s” represents the sample standard deviation.

s = the square root of 25.6 = 5.06

Standard DeviationThe population standard deviation can be estimated by calculating the sample standard deviation. The formulas for sample and population look similar except that the sample standard deviation formula uses n – 1 instead of n in the denominator. This is to account for the possibility of greater variability of data in the population than what was seen in the sample.

Standard DeviationWhen n-1 is used in the denominator, the result is a larger number. So the calculated value of the sample standard deviation will be larger than the population standard deviation. As sample sizes get larger, the difference gets smaller. The use of n-1 is known as Bessel’s correction. SOL A.9 used the population standard deviation with n in the denominator.

Interpreting Standard Deviation• Standard deviation is a measure of the typical

amount an entry deviates from the mean.• The more the entries are spread out, the greater

the standard deviation.

Interpreting Standard DeviationEmpirical Rule (68 -95-99.7 Rule)

For data with a (symmetric) bell-shaped distribution, the standard deviation has the following characteristics:•About 68% of the data lie within one standard deviation of the mean.•About 95% of the data lie within two standard deviations of the mean.•About 99.7% of the data lie within three standard deviations of the mean.

Empirical Rule (68 – 95 – 99.7 Rule)

3x s x s 2x s 3x sx s x2x s

68% within 1 standard deviation

34% 34%

99.7% within 3 standard deviations

2.35% 2.35%

95% within 2 standard deviations

13.5% 13.5%

Example: Using the Empirical Rule

In a survey conducted by the National Center for Health Statistics, the sample mean height of women in the United States (ages 20-29) was 64.3 inches, with a sample standard deviation of 2.62 inches. Estimate the percent of the women whose heights are between 59.06 inches and 64.3 inches.

Example: Using the Empirical Rule

• Because the distribution is bell-shaped, you can use the Empirical Rule.

34% + 13.5% = 47.5% of women are between 59.06 and 64.3 inches tall.

Chebychev’s Theorem• The portion of any data set lying within k standard

deviations (k > 1) of the mean is at least:

• k = 2: In any data set, at least

of the data lie within 2 standard deviations of the

mean.• k = 3: In any data set, at least

of the data lie within 3 standard deviations of the

mean.

2

11

k

2

1 31 or 75%

2 4

2

1 81 or 88.9%

3 9

Using Chebychev’s TheoremThe age distribution for Florida is shown in the histogram. Apply Chebychev’s Theorem to the data using k = 2. What can you conclude?

Using Chebychev’s Theorem

k = 2: μ – 2σ = 39.2 – 2(24.8) = – 10.4 (use 0 since age can’t be negative)

μ + 2σ = 39.2 + 2(24.8) = 88.8

At least 75% of the population of Florida is between 0 and 88.8 years old.

z-ScoresA z-score, also called a standard score, is a measure of the position derived from the mean and standard deviation of the data set. In Algebra I, the z-score will be used to determine how many standard deviations an element is above of below the mean of the data set. It can also be used to determine the value of the element, given the z-score of an unknown element and the mean and standard deviation of a data set.

The Standard Score (z-Score)

Represents the number of standard deviations a given value x falls from the mean μ.

z-Scores

The z-score will be positive if the element lies above the mean and negative if it lies below the mean. A z-score is calculated by subtracting the mean of the data set from the element and dividing the result by the standard deviation of the data set.

*z-Scores*

z-score (z) =

where x represents an element of the data set, m represents the mean of the data set, and s represents the standard deviation of the data set.

z-Scoresz-score of 5 = (5 - 12)/5.06 = -1.38z-score of 6 = (6 - 12)/5.06 = -1.19z-score of 8 = (8 - 12)/5.06 = -.79z-score of 10 = (10 - 12)/5.06 = -.40z-score of 13 = (13 - 12)/5.06 = .20z-score of 15 = (15 - 12)/5.06 = .59z-score of 17 = (17 - 12)/5.06 = .99z-score of 18 = (18 - 12)/5.06 = 1.19z-score of 20 = (20 - 12)/5.06 = 1.58

z-ScoresSuppose you had the misfortune to have an Algebra test and a history test on the same day. Why you got your tests back, here is the information given to you regarding your performance and the performance of the class on these exams. Which test did you do better on?

AlgebraHistory

Your score 82 93Mean 71.06 85.43Stan. Dev 10.32 18.91

z-ScoresSuppose the mean and standard deviation on an algebra test were given as 72 and 12, respectively.

Susan’s z-score on the test was 2.34. Was Susan’s test score above or below the mean? How do you know?

David’s z-score on the same test was -1.25. Was David’s test score above or below the mean? How do you know?


Dakota had a z-score of 0.08 on the test. What does this z-score tell you about Dakota’s test score relative to the mean?


If the z-score for Susan was 2.34, for David was -1.25, and for Dakota was 0.08, find their actual test scores.


Rebecca made an 80 on the biology test. Find her z-score.

Interpretation of Descriptive Statistics

Data set 1 Data set 2

Number of Basketball Players Recorded Once Each Day from April 1-14

Number of Basketball Players Recorded Once Each Day from April 15-28

7 7

6 6

5 5

4 4

3 3

2 2

1 1

0 0

1-10 11-20 21-30 31-40 41-50 51-60 61-70

Number of Players

Mean = 45.0 Variance = 106.1 Standard Deviation = 10.3 Mean Absolute Deviation = 9.1

1-10 11-20 21-30 31-40 41-50 51-60 61-70

Number of Players

Mean = 45.0 Variance = 420.3 Standard Deviation = 20.5 Mean Absolute Deviation = 16

Interpretation of Descriptive Statistics

Maya represented the heights of boys in Mrs. Constantine’s and Mr. Kluge’s classes on a line plot and calculated the mean and standard deviation.

Heights of Boys in Mrs. Constantine’s and Mr. Kluge’s Classes (in inches)

x

x

x x x

x x x x x x

x x x x x x x x x x64 65 66 67 68 69 70 71 72 73

Mean = 68.4 Standard Deviation = 2.3

Note: In this problem, a small, defined population of the boys in Mrs. Constantine’s and Mr.

Kluge’s classes is assumed.

Margie Mason

M. Mason, College of William and Mary Tidewater Team

A Look at Means, Variances, Standard Deviations, and z-Scores Dr. Margie Mason College of William...

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