STATS (beginners)

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Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Measures of Center

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Key Concept

When describing, exploring, and comparing data sets, these characteristics are usually extremely important: center, variation, distribution, outliers, and changes over time.

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What are we going to learn?

• Measures of Center

Mean, Median, Mode

• Range

• Standard Deviation

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Arithmetic Mean

(Mean)

You hear people sometimes say average

the measure of center obtained by adding the values and dividing the total by the number of values

Definition

N

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Notation

 denotes the sum of a set of values (sigma)

x is the variable usually used to represent the individual data values.

n represents the number of values in a sample (because it is lower case)

N represents the number of values in a population (because it is upper case)

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Notation

µ is pronounced ‘mew’ and denotes the mean of all values in a population

is pronounced ‘x-bar’ and denotes the mean of a set of sample values

Symbol for the sample mean

Symbol for the population mean

This equation reads the mean equals the sum of each score divided by n

x =

n

 x

x

N

µ =

 x

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Let’s Practice Some Mean Exercises
Get out a calculator

Determine the mean for each of the data sets:

3, 5, 7

6, 5, 6, 2, 1

3, 21, 16, 9, 13

7, 3, 6, 3, 2

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Let’s Practice Some Mean Exercises
Get out a calculator

Determine the mean for each of the data sets:

3, 5, 7 = 3 + 5+ 7 = 15/3 =

b) 6, 5, 6, 2, 1 = 6 + 5 + 6 + 2 + 1 = 20/5

3, 21, 16, 9, 13 = 3 + 21 + 16 + 9 + 13 = 62/5

7, 3, 6, 3, 2 = 21/5

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Answers: a) 5; b) 4 c) 12.4 ;d) 4.2

Don’t assume you know this. Please practice with your calculator.

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Let’s Try this Exercise

The table shows daily sales in the first 2 weeks of July in Windsor Mill at the sandwhich shop.

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Let’s Try this Exercise

What is the mean sales for the data?

56.7

59

60

63.5

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Answer

What is the mean sales for the data?

56.7

59

60

63.5

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Definitions

Mode

the value that occurs most frequently

Mode is not always unique

A data set may be:

Bimodal (two modes)

Multimodal (many modes)

No Mode

Mode is the only measure of central tendency that can be used with nominal data

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a. 5.40 1.10 0.42 0.73 0.48 1.10

b. 27 27 27 55 55 55 88 88 99

c. 1 2 3 6 7 8 9 10

Mode - Examples

Mode is 1.10

Bimodal - 27 & 55

No Mode

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Let’s Practice Finding the Mode

Find the mode in the following data sets:

17, 19, 19, 4, 19, 26, 3, 21, and 19

3, 4, 4, 5, 2, 1, 3, 0, 0, 3

0, 0, 2, 1, 1 5, 6, 3, 7, 0, 3, 1

Answers: a) 19; b) 3 c) 0 and 1

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Definitions

Median

the middle score

• is not affected by an extreme value

• often denoted by x (pronounced ‘x-tilde’)

~

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Finding the Median Steps

Step 1 – Place the data in an ordered array (from lowest to highest)

Step 2 – Determine N

Step 3a - If the number of scores (N) is odd, the median is the number located in the exact middle of the list.

Step 3b - If the number of scores (N) is even, the median is found by computing the mean of the two middle numbers.

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5.40 1.10 0.42 0.73 0.48 1.10 0.66

0.42 0.48 0.66 0.73 1.10 1.10 5.40

(in order - odd number of values)

exact middle MEDIAN is 0.73

5.40 1.10 0.42 0.73 0.48 1.10

0.42 0.48 0.73 1.10 1.10 5.40

0.73 + 1.10

2

(in order - even number of values – no exact middle

shared by two numbers)

MEDIAN is 0.915

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Let’s Practice Finding the Median

Find the median for the following data set:

• A) 92, 2, 90, 86, 96
• B) 12.5, 10.5, 24.5, 26, 15, 21
• C) 250, 240, 188, 198, 240, 230, 210, 225

Answers: A) 90; B) 18; C) 227.5

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Let’s Try this Exercise

The table shows daily sales in the first 2 weeks of July in Windsor Mill at the sandwhich shop.

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Let’s Try this Exercise

What is the median for the data?

50

55

58

60

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Answer

What is the median for the data?

50

55

58

60

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Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Q1, Q2, Q3

divide ranked scores into four equal parts

Quartiles

25%

25%

25%

25%

Q3

Q2

Q1

(minimum)

(maximum)

(median)

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Definition

• Q1 (First Quartile) separates the bottom 25% of sorted values from the top 75%.

• Q2 (Second Quartile) same as the median; separates the bottom 50% of sorted values from the top 50%.

• Q1 (Third Quartile) separates the bottom 75% of sorted values from the top 25%.

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Here are the steps…

• Step 1 – Put the data into an ordered array (from lowest to highest)
• Step 2 – Find the median for the entire data set (Q2)
• Step 3 – Find the median for the first half of the data set (Q1)
• Step 4 – Find the median for the 2nd half of the data set (Q3)

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Step 1

0, 11, 22, 2, 3, 6, 21, 32, 2, 2, 9, 17, 21, 14, 19

0, 2, 2, 2, 3, 6, 9, 11, 14, 17, 19, 21, 21, 22, 32

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Step 2

0, 2, 2, 2, 3, 6, 9, 11, 14, 17, 19, 21, 21, 22, 32

Q2

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To find the top/bottom half, put a line through Q2

0, 2, 2, 2, 3, 6, 9, 11, 14, 17, 19, 21, 21, 22, 32

The top half of the data becomes 0, 2, 2, 2, 3, 6, 9.

The bottom half of the data becomes 14, 17, 19, 21, 21, 22, 32

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Step 3

0, 2, 2, 2, 3, 6, 9

Q1

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Step 4

14, 17, 19, 21, 21, 22, 32

Q3

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So the 3 data points are:

Q1 = 2; Q2 = 11; Q3 = 21

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Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

• Interquartile Range (or IQR): Q3 - Q1

Some Other Statistics

• Five Number Summary:

Min, Q1, Q2, Q3, Max

2

Q3 - Q1

• Semi-interquartile Range:

2

Q3 + Q1

• Midquartile:

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Let’s Try this Exercise

The table shows daily sales in the first 2 weeks of July in Windsor Mill at the sandwhich shop.

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Let’s Try this Exercise

What is the five-number summary for the data?

a) 26, 46, 55, 86, 96

b) 25, 45, 55, 85, 96

c) 26, 46, 56, 85, 96

d) 25, 46, 55, 86, 96

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Answer

What is the five-number summary for the data?

a) 26, 46, 55, 86, 96

b) 25, 45, 55, 85, 96

c) 26, 46, 56, 85, 96

d) 25, 46, 55, 86, 96

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Midrange

the value midway between the maximum and minimum values in the original data set

Definition

Midrange =

maximum value + minimum value

2

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Carry one more decimal place than is present in the original set of values.

Round-off Rule for

Measures of Center

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Best Measure of Center

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Skewness

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Recap

In this section we have discussed:

• Types of measures of center

Mean

Median

Mode

• Best measures of center

• Skewness

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Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Measures of Variation

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Key Concept

Because this section introduces the concept of variation, which is something so important in statistics, this is one of the most important sections in the entire book.

Place a high priority on how to interpret values of standard deviation.

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Let’s try some basic computations

Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Determine Σxy and Σx2

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Let’s try some basic computations

Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Determine Σxy

Determine the sum of x times y

12 + 2 + 20 +42 = 76

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Let’s try some basic computations

Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Determine Σx2

Determine the sum of x squared

4 + 2 + 16 + 49 = 71

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Now try one on your own

Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Determine Σxy and Σx2

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Did you get?

Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Determine Σxy and Σx2

90 and 103

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Definition

The range of a set of data is the difference between the maximum value and the minimum value.

Range = (maximum value) – (minimum value)

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Definition

The standard deviation of a set of sample values is a measure of variation of values about the mean.

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Sample Standard
Deviation Formula

Let’s read this formula aloud. S which stands for standard deviation is equal to the square root of the sum of each score minus the mean squared divided by n-1.

 (x - x)2

n - 1

s =

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Standard Deviation -

Important Properties

• The standard deviation is a measure of variation of all values from the mean.

• The value of the standard deviation s is usually positive.

• The value of the standard deviation s can increase dramatically with the inclusion of one or more outliers (extreme scores).

• The units of the standard deviation s are the same as the units of the original data values.

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Population Standard
Deviation

2

 (x - µ)

N

 =

This formula is similar to the previous formula, but instead, the population mean and population size are used.

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Definition

• The variance of a set of values is a measure of variation equal to the square of the standard deviation.

• Sample variance: Square of the sample standard deviation s

• Population variance: Square of the population standard deviation

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Variance - Notation

standard deviation squared

s



2

2

}

Notation

Sample variance

Population variance

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Round-off Rule
for Measures of Variation

Carry one more decimal place than is present in the original set of data.

Round only the final answer, not values in the middle of a calculation.

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Let me now show you how to computer s (standard deviation)

• I do it very simply…with a chart
• Let’s consider that we want to find the standard deviation for the following data set: 5, 4, 2, 3, 6

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Let me now show you how to computer s (standard deviation)

• Let’s consider that we want to find the standard deviation for the following data set: 5, 4, 2, 3, 6
• Step 1 – Place each score in the chart and sum.
• Step 2 – Find the mean.
• Step 3 – Identify N



Total this row

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Let me now show you how to computer s (standard deviation)

• Let’s consider that we want to find the standard deviation for the following data set: 5, 4, 2, 3, 6
• Step 1 – Place each score in the chart and sum.



Total this row

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Let me now show you how to computer s (standard deviation)

• Let’s consider that we want to find the standard deviation for the following data set: 5, 4, 2, 3, 6
• Step 2 – Find the mean



Total this row

20/5 = 4

Mean = 4

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Let me now show you how to computer s (standard deviation)

• Let’s consider that we want to find the standard deviation for the following data set: 5, 4, 2, 3, 6
• Step 3 – Identify N



Total this row

Mean =4

N = 5

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Let me now show you how to computer s (standard deviation)

• Let’s consider that we want to find the standard deviation for the following data set: 5, 4, 2, 3, 6
• Step 4 – Now subtract each score from the mean



Total this row

Mean =4

N = 5

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Let me now show you how to computer s (standard deviation)

• Let’s consider that we want to find the standard deviation for the following data set: 5, 4, 2, 3, 6
• Step 5 – Total this column. The total should almost always be 0.



Total this row

Mean =4

N = 5

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Let me now show you how to computer s (standard deviation)

• Let’s consider that we want to find the standard deviation for the following data set: 5, 4, 2, 3, 6
• Step 6 – Square each score.



Total this row

Mean =4

N = 5

Squared means the number times itself. On your calculator is an x squared button.

Do not put negative numbers in your calculator. The last column will always be positive.

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Let me now show you how to computer s (standard deviation)

• Let’s consider that we want to find the standard deviation for the following data set: 5, 4, 2, 3, 6
• Step 7 – Total this column.



Total this row

Mean =4

N = 5

Squared means the number times itself. On your calculator is an

This total becomes the numerator in the formula.

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Sample Standard
Deviation Formula

10

Now the formula comes to life and becomes…

Original Formula

n

 (x - x)2

n - 1

s =

5 - 1

s =

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Sample Standard
Deviation Formula

10

10

Watch how the formula works itself out…

5 - 1

s =

4

s =

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Sample Standard
Deviation Formula

10

2.5

Watch how the formula works itself out…

4

s =

s =

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Sample Standard
Deviation Formula

2.5

Compute the square root of 2.5

s = 1.58

Voila!

s =

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Sample Standard Deviation Formula

• I know that was a lot to take in at once; however, let’s try one now by yourself. Just take your time. Standard deviation takes practice to get perfect, but with a little practice you will get it.
• Take your time and check your work. DON’T RUSH!

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Sample Standard Deviation Formula

• Find s for the following data sets:
• A) 4, 2, 3
• B) 9, 6, 7, 2
• C) 2, 6, 1
• D) 498, 500, 503, 498, 501

Answers: a) 1; b) 2.94; c) 2.646; d) 2.12

Do not move forward unless you have the answers.

I mean it!

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Definition

Empirical (68-95-99.7) Rule

For data sets having a distribution that is approximately bell shaped, the following properties apply:

• About 68% of all values fall within 1 standard deviation of the mean.

• About 95% of all values fall within 2 standard deviations of the mean.

• About 99.7% of all values fall within 3 standard deviations of the mean.

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The Empirical Rule

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The Empirical Rule

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The Empirical Rule

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What does standard deviation mean?

• Standard deviation is a measure which talks about how the scores scatter around the mean.
• If s is low, the scores, in general, are closer to the mean.
• If s is high, the scores, in general, are farther from the mean.
• Think of yourself as the mean and you are holding jellybeans in your clinched fist. You open your hand and drop them at your feet.
• Some jellybeans will be right by your feet and a few may scatter a few feet away. S would be low. In general, the jellybeans are staying close to you.
• Think of yourself again as the mean and you are holding jellybeans in your clinched fist. This time, you open your hand and toss the jellybeans in front of you. S will be high. In general, some jellybeans are near you, but more of them are away from you.

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What else does standard deviation mean?

• There are 3 steps on the standard deviation: 68%, 95%, and 99%
• How many steps? Three
• And what are they? 68% 95% 99%

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Here is standard deviation at work

• Imagine the first statistics exam. The mean is 80 and the standard deviation for our class is 5.

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The Empirical Rule

80

mean

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The Empirical Rule

80 + 5 = 85

s = 5

85

75

80 - 5 = 75

Finding: 68% of the class scored between 75 and 85 on the first STAT exam.

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The Empirical Rule

75 - 5 = 70

85 + 5 = 90

70

90

Finding: 95% of the class scored between 70 and 90 on the first STAT exam.

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70 - 5 = 65

90 + 5 = 95

65

95

Finding: 99% of the class scored between 65 and 95 on the first STAT exam.

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Definition

An outlier is a value that is located very far away from almost all of the other values.

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Important Principles

• An outlier can have a dramatic effect on the mean.
• An outlier can have a dramatic effect on the standard deviation.
• An outlier can have a dramatic effect on the scale of the histogram so that the true nature of the distribution is totally obscured.

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Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Key Concept

This section introduces measures that can be used to compare values from different data sets, or to compare values within the same data set. The most important of these is the concept of the z score.

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z Score (or standardized score)

the number of standard deviations that a given value x is above or below the mean

Definition

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Sample

Round z to 2 decimal places

Measures of Position z score

z =



Population

x - µ

z =

x - x

s

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Interpreting Z Scores

Whenever a value is less than the mean, its corresponding z score is negative

Ordinary values: z score between –2 and 2 Unusual Values: z score < -2 or z score > 2

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Let’s compute some z-scores

• A normal distribution has a mean of 50 and a standard deviation of 8.
• Compute the z-score for
• A) X = 56
• B) X = 43
• C) X = 68

Answers: a) 0.75; b) -0.875; c) 2.25

56-50

8

43-50

8

68-50

8

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For a set of data, the 5-number summary consists of the minimum value; the first quartile Q1; the median (or second quartile Q2); the third quartile, Q3; and the maximum value.

• A boxplot ( or box-and-whisker-diagram) is a graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile, Q1; the median; and the third quartile, Q3.

Definitions

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Boxplots

A boxplot is another way to display data.

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Boxplots - cont

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Boxplots - cont

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Boxplots - cont

Is this distribution left or right skewed?

Right-skewed

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Practice Exercise for Chapter 3

The amounts of fuel oil (in gallons) consumed for a sample of both newer and older homes is reported below:

Newer homes: 600, 590, 605, 600, 603, 610, 597, 595

Older homes: 810, 750, 790, 800, 850

Determine the standard deviation and variance for newer homes

Determine the standard deviation and variance for older homes

Which homes are using less oil and why? (hint: use the 3 steps rule)

Compute the z score for the new home that consumes 597 gallons.

Compute the z score for the old home that consumes 790 gallons.

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• Send me your work directly at [email protected] for the Practice Exercise answers.

Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.