Objective To understand measures of central tendency and use them to analyze data.

Objective

To understand measures of central tendency and use

them to analyze data.

Measures of Central TendencyMean, Median and Mode

Mean – the average Sum of the data items Total number of data items

Used to describe the middle of a set of data that does not have outliers (data values that are much higher or lower than other values in the set.)

Find the Mean

Q: 4, 5, 8, 7

A: 6

Median: 6

Q: 4, 5, 8, 1000

A: 254.25

Median: 6.5

Median

The middle value in a set of data where the numbers are arranged in order.

Used to describe the middle of a set of data that does have outliers.

If the data has an even number of items the median is the average of the middle two numbers.

Median

Find the Median

4 5 6 6 7 8 9 10 12 Find the Median

5 6 6 7 8 9 10 12 Find the Median

5 6 6 7 8 9 10 100,000

Mode

The data item that occurs the most times

Can be used when data is not numeric

Can have one, two or more modes.

Used to choose the most popular outcome.

Mode

Most Common Outcome

Male Female

Measures of Central TendencyMode

The most common observation in a group of scores. Distributions can be unimodal, bimodal, or multimodal.

If the data is categorical (measured on the nominal scale) then only the mode can be calculated.

The most frequently occurring score (mode) is Vanilla.

0

5

10

15

20

25

30

Vanilla

Choco

late

Strawbe

rry

Neapo

litan

Butte

r Pec

an

Rocky

Roa

d

Fudg

e Ripp

le

fFlavor f

Vanilla 28

Chocolate 22

Strawberry 15

Neapolitan 8

Butter Pecan 12

Rocky Road 9

Fudge Ripple 6

Range

The difference between the least and greatest data values.

Find the range of: 2, 34, 55, 22, 4, 7, 84, 55, 77

Summarizing Distributions

Two key characteristics of a frequency distribution are especially important when summarizing data or when making a prediction from one set of results to another:

Central Tendency What is in the “Middle”? What is most common? What would we use to predict?

Dispersion How Spread out is the distribution? What Shape is it?

Measures of Variability

Central Tendency doesn’t tell us everything

Dispersion/Deviation/Spread tells us a lot about how a variable is distributed.

We are most interested in Standard Deviations (σ)

Standard Deviation

A measure of dispersion that describes the typical difference or deviation between the mean and a data value.

Subtract the mean from each data value and square this difference. Do this for each data value and add the answers together. Divide the sum by the number of data items then take the square root.

2( )iX Xs

N

Find the standard deviation for the following test scores

98, 72, 55, 88, 69, 92, 77, 89, 94, 70

Line Plot

Used to show frequency.

Tally Chart

Used to show frequency.

Stem and Leaf Plot

Used to organize data.

Easy to see the mode!

Practice!

Use the following data to make a stem and leaf plot. Find the mean, median, mode and range of the data.

18, 35, 28, 15, 36, 72, 14, 55, 62, 45, 80, 9, 72, 66, 28, 20, 51, 44, 28

Mean 40.95 Median 36 Mode 28 Range 71

A histogram is a bar graph that shows the frequency of data within equal intervals. There is no space between the bars in a histogram.

Course 2

1-4 Bar Graphs and Histograms

Histograms

The histogram is a tool for presenting the distribution of a numerical variable in graphical form.

For example, suppose the following data is the number of hours worked in a week by a group of nurses:

42 47 43 26 30 42 28 42 50 3938 35 37 48 39 36 45 41 72 5343 37 42 48 40 35 39 30 47 38

Histograms

Hours worked in the week

12

10

8

6

4

2

0

25 30 35 40 45 50 55 60 65 70 75

The verticalaxis isfrequency. So,for example,there are twonurses whoworked from25 to less than 30 hours that week.

The data values are grouped in intervals of width five hours. The first interval includes the values from 25 to less than 30 hours. The second interval includes values from 30 to less than 35 and so on. The intervals are shown on the horizontal axis.

26

28

30

30

35

35

36

37

37

38

38

39

39

39

40

41

42

42

42

42

43

43

45

47

47

48

48

50

53 72

These data are displayed in the following histogram:

Histograms

Hours worked in the week

12

10

8

6

4

2

0

25 30 35 40 45 50 55 60 65 70 75

The choice of interval width will affect the appearance of the histogram.

Hours worked in the week

20

10

0

25 35 45 55 65 75

Hours worked in the week

6

5

4

3

2

1

0

26 30 34 38 42 46 50 54 58 62 66 70 74

To the right is the same data presented in a histogram of interval width 10.

And here it is again, to the right, presented in a histogram of interval width 2.

The table below shows the number of hours students watch TV in one week. Make a histogram of the data.

Additional Example 3: Making a Histogram

Step 1: Make a frequency table of the data. Be sure to use equal intervals.

Course 2

1-4 Bar Graphs and Histograms

6 ///

7 //// ////

8 ///

9 ////

1 //

2 ////

3 //// ////

4 //// /

5 //// ///

Number of Hours of TV

1–3

FrequencyNumber of Hours of TV

15

4–6 17

7–9 17

Additional Example 3 Continued

Step 2: Choose an appropriate scale and interval for the vertical axis. The greatest value on the scale should be at least as great as the greatest frequency.

Course 2

1-4 Bar Graphs and Histograms

1–3

FrequencyNumber of Hours of TV

15

4–6 17

7–9 17

20

16

12

8

4

0

Additional Example 3 Continued

Step 3: Draw a bar graph for each interval. The height of the bar is the frequency for that interval. Bars must touch but not overlap.

Course 2

1-4 Bar Graphs and Histograms

1–3

FrequencyNumber of Hours of TV

15

4–6 17

7–9 17

20

16

12

8

4

0

Additional Example 3 Continued

Step 4: Label the axes and give the graph a title.

Course 2

1-4 Bar Graphs and Histograms

1–3

FrequencyNumber of Hours of TV

15

4–6 17

7–9 17

20

16

12

8

4

01–3 4–6 7–9

Hours of Television Watched

Frequ

ency

Hours

The table below shows the number of hats a group of students own. Make a histogram of the data.

Try This: Example 3

Step 1: Make a frequency table of the data. Be sure to use equal intervals.

Course 2

1-4 Bar Graphs and Histograms

1–3

FrequencyNumber of Hats Owned

12

4–6 18

7–9 24

1 //

2 ////

3 //// /

4 //// /

5 //// ///

6 ////

7 //// /

8 //// ////

9 //// ////

Number of Hats Owned

Frequency

Try This: Example 3

Step 2: Choose an appropriate scale and interval for the vertical axis. The greatest value on the scale should be at least as great as the greatest frequency.

Course 2

1-4 Bar Graphs and Histograms

1–3

FrequencyNumber of Hats Owned

12

4–6 18

7–9 24

30

25

20

15

10

5

0

Try This: Example 3

Step 3: Draw a bar graph for each interval. The height of the bar is the frequency for that interval. Bars must touch but not overlap.

Course 2

1-4 Bar Graphs and Histograms

1–3

FrequencyNumber of Hats Owned

12

4–6 18

7–9 24

30

25

20

15

10

5

0

Try This: Example 3

Step 4: Label the axes and give the graph a title.

Course 2

1-4 Bar Graphs and Histograms

1–3

FrequencyNumber of Hats Owned

12

4–6 18

7–9 24

30

25

20

15

10

5

01–3 4–6 7–9

Number of Hats Owned

Frequ

ency

Number of Hats

Lesson Quiz: Part 1

1. The list shows the number of laps students ran one day. Make a histogram of the data.

4, 7, 9, 12, 3, 6, 10, 15, 12, 5, 18, 2, 5, 10, 7, 12, 11, 15

Course 2

1-4 Bar Graphs and Histograms

Nu

mb

er

of

Stu

den

ts Number of Laps Run

10–1

4

0–4

5– 9

8

6

4

2

0

15–1

9

Number of Laps

Normally Distributed Curve

Characteristics of the Normal DistributionIt is symmetrical -- Half the cases are to one side of

the center; the other half is on the other side.The distribution is single peakedMost of the cases will fall in the center portion of the

curve and as values of the variable become more extreme they become less frequent, with “outliers” at each of the “tails” of the distribution few in number.

The Mean, Median, and Mode are the same.Percentage of cases in any range of the curve can be

calculated.

Skewed Distributions

The total area under the normal curve is 1.

About 68% of the area lies within 1 standard deviation of the mean.

About 95% of the area lies within 2 standard deviations of the mean.

About 99.7 % of the area lies within 3 standard deviations of the mean.

Why can’t the mean tell us everything?

Mean describes Central Tendency, what the average outcome is.

We also want to know something about how accurate the mean is when making predictions.

The question becomes how good a representation of the distribution is the mean? How good is the mean as a description of central tendency -- or how good is the mean as a predictor?

Answer -- it depends on the shape of the distribution. Is the distribution normal or skewed?

DispersionOnce you determine that the variable of interest is

normally distributed, the next question to be asked is about its dispersion: how spread out are the scores around the mean.

Dispersion is a key concept in statistical thinking.

How much do the scores deviate around the Mean? The more “bunched up” around the mean the better your ability to make accurate predictions.

How well does the mean represent the scores in a distribution? The logic here is to determine how much spread is in the scores. How much do the scores "deviate" from the mean? Think of the mean as the true score or as your best guess. If every X were very close to the Mean, the mean would be a very good predictor.

If the distribution is very sharply peaked then the mean is a good measure of central tendency and if you were to use the mean to make predictions you would be right or close much of the time.