Common Core Math I Unit 2: One-Variable Statistics Measures of Center and Spread

Common Core Math I Unit 2: One-Variable Statistics

Measures of Center and Spread

Think About the SituationDiscuss the following with your partner or group. Write

your answers on your own paper. Be prepared to share your answers with the class.

• What do you think is the typical number of letters in the full names (first and last) of your classmates?

• What data do you need to collect and how would you collect it?

• How would you organize and represent your data?• If a new student joined your class today, how might

you use your results to predict the length of that student’s name?

Investigation 1: Dotplots vs. Histograms

Describing Data

Two ways to describe data:• Graphically

Dot plotHistogram– Boxplot

• Numerically– Measures of Center– Measures of Spread

Describing Data Numerically• Measures of Center –

mean, median

• Measures of Spread – range, interquartile range, standard deviation

S-ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

Measures of CenterWhat is the typical value?

Investigation 2: Finding the Median

Investigation 3: Experimenting With the Median

Name Number of Letters

Pete Thomas 10

Shaquana Smith

Stewart Hughes

Huang Mi

Richard Lewis

Virginia Bates

Angel Mendoza

Mary Wall

Danielle Duncan

Will Jones

Ana Romero

Jana Turner

HuangMi

7letters

Describing Data Numerically• Measures of Center –

mean, median

• Measures of Spread – range, interquartile range, standard deviation

S-ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

Think About the SituationUse cubes to make stacks representing each household. Use the stacks to answer the following questions.What is the median of these data?

Adapted from Data About Us, Connected Mathematics 2, Grade 6

Ossie

Leon

Gary

Ruth

Paul

Arlene

Finding the Mean

Adapted from Data About Us, Connected Mathematics 2, Grade 6

Ossie

Leon

Gary

Ruth

Paul

Arlene

Make stacks all the same height by moving cubes.

Finding the Mean• How many cubes are in each stack?• By leveling out the stacks to make them equal

height, you have found the average, or mean, number of people in a household. What is the mean number of people per household?

Adapted from Data About Us , Connected Mathematics 2, Grade 6

Ossie

Leon

Gary

Ruth

Paul

Arlene

Investigation 1: Finding the Mean

Student Name

Number of People

in Household

Reggie 6

Tara 4

Brendan 3

Felix 4

Hector 3

Tonisha 4

• How many people are in the six households altogether? Explain.

• What is the mean number of people per household for this group? Explain how you got that number.

• How does the mean for this group compare to the mean of the first group?

• What are some ways to determine the mean number of a set of data other than using cubes? How do these methods relate to the method of using the cubes?

Mean as a Fair Share

• Mary – 7 cookies

• Tom – 6 cookies

• Julio – 2 cookies

Mean as a Fair Share

• How many cookies will each child get if they each get an equal share?

Mean as a Balance Point

2 43 5 6 7

Mean as a Balance Point

Mean as a Balance Point

Mean as a Balance Point

Mean as a Balance Point

Mean as a Balance Point

Mean as a Balance Point

Mean as Balance Point

Mean as a Balance Point

Thinking About the Situation

The Formula

n

xx

x

valuesdata ofnumber

valuesdata of sum

Investigation 2: Data with the Same Mean1) Find two new data sets for six households that each have a mean of 4 people

per household. Use cubes to show each data set. Then make dotplots from the cubes.

2) Find two different data sets for seven households that each has a mean of 4 people per household. Use cubes to show each data set. Then make dotplots from the cubes.

3) A group of seven students find that they have a mean of 3 people per household. Find a data set that fits this description. Then make a dot plot for this data.

4) A group of six students has a mean of 3.5 people per household. Find a data set that fits this description. Then make a dot plot for this data.

5) How can the mean be 3 ½ people when “half” a person does not exist?

6) How can you predict when the mean number of people per household will not be a a whole number?

Investigation 3: Using the Mean

Investigation 3: Using the Mean1) Find the following:

a) the total number of studentsb) the total number of movies watchedc) the mean number of movies watched

2) A new value is added for Carlos, who was home last month with a broken leg. He watched 31 movies.a) How does the new value change the distribution on the

histogram?b) Is this new value an outlier? Explain.c) What is the mean of the data now?d) Compare the mean from question 1 to the new mean.

What do you notice? Explain. e) Does this mean accurately describe the data? Explain.

3) Data for eight more students is added.a) Add these values to the list in your calculator.

How do these values change the distribution on the histogram?

b) Are any of these new values outliers?c) What is the mean of the data now?

5 10 15 20 250

1

2

3

4

5

6

7

8

9

Movies Watched

Number of Movies

Num

ber o

f Stu

dent

s

How do I know which measure of central tendency to use?

http://regentsprep.org/REgents/math/ALGEBRA/AD2/measure.htm

Investigation 4: Mean vs. Median

The heights of Washington High School’s basketball players are: 5 ft 9in, 5 ft 4in, 5 ft 7 in, 5ft 6 in, 5 ft 5 in, 5 ft 3 in, and 5 ft 7 in. A student transfers to Washington High and joins the basketball team. Her height is 6 ft 10in.

Discuss and solve in your groups!

Mean vs. Median

http://www.stat.tamu.edu/~west/ph/meanmedian.html

Mean vs. MedianMound-shaped

and symmetrical (Normal)

Skewed Left

Skewed Right

Ticket out the DoorWhat happens to the mean of a data set when you add one or more data values that are outliers? Explain.

What happens to the mean of a data set when you add values that cluster near one end of the original data set? Explain.

Explain why you think these changes might occur.

Describing Data

Two ways to describe data:• Graphically

Dot plotHistogram– Boxplot

• Numerically– Measures of Center: Median and Mean– Measures of Spread

Measures of Spread

• Range

• Interquartile Range (IQR)

• Standard Deviation

How much do values typically vary from the center?

Thinking about the SituationConsider the following test scores:

Who is the best student?How do you know?

Student Test 1 Test 2 Test 3 Test 4

Johnny 65 82 93 100

Will 82 86 89 84

Anna 80 99 73 88

Measures of Spread

One-Variable:RangeInterquartile Range (IQR)Mean Absolute Deviation (MAD)Standard Deviation

How much do values typically vary from the center?

Score MeanDeviation from

the Mean

Anna

Test 1

Test 2

Test 3

Test 4

Investigation 1: Deviation from the Mean

x xx x

So what exactly is deviation?

0 1 2 3 4 5 6 7 8 9 10 11

+3

+5

-1

-3

-4

(-4) + (-3) + (-1) = -8 (+5 ) + (+3) = +8

x

Measure of SpreadSum of deviations = (-4)+(-3)+(-1)+(+5)+(+3)

Average of the deviations= = 0

An average deviation of zero means that there is no variability!

Houston, we have a

problem!

5

0

How can we fix our problem?• Take the absolute value of each distance/deviation and then find the

average

• So the average distance or deviation from the mean is about 3 points (above or below).

• This is called the Mean Absolute Deviation, or MAD

2.35

16 average

165 3 1 3 4

mean thefrom deviations theof

valueabsolute theof sum

Mean Absolute Deviation

0 1 2 3 4 5 6 7 8 9 10 11

+3.2-3.2

x

How can we fix our problem?Investigation 2: Mean Absolute Deviation

Deviation from the Mean

Absolute Deviation from the Mean

Johnny

Test 1

Test 2

Test 3

Test 4 Will

Test 1

Test 2

Test 3

Test 4 Anna

Test 1

Test 2

Test 3

Test 4

Square each deviation and then find the average of the squared deviations.

This is called the standard deviation.

125

60 deviation squared average

602591916

5313)4(

deviations squared of sum22222

4.312

squaring thedo"-un" root to square theThen take

Back to Johnny, Will and Anna . . .

Investigation 3: Calculating the Standard Deviation

Student Test 1 Test 2 Test 3 Test 4

Johnny 65 82 93 100

Will 82 86 89 84

Anna 80 99 73 88

Who is the best student?How do you know?

Student Test 1 Test 2 Test 3 Test 4 TestAverage

Standard Deviation

Johnny 65 82 93 100 85 13.2

Will 82 86 89 84 85 2.2

Anna 80 99 73 88 85 9.7

Standard Deviation

n

x 2)(

s vs. What’s the difference?

n

x

2)(

1

)( 2

n

xxs

population sample