Day 1 Notes – 11.1 Data Distributions/Measures of Central Tendency

Data Distribution

Let’s construct class

HISTOGRAMS

With Post-Its

4 Main Histogram Shapes

What are examples of data that might be...

a. Skewed Left? B. Skewed Right C. Symmetrical?

Data Shape: Range:

(diff between largest

and smallest values)

Class Height

Class Ages

Number of Siblings

Objectives: 1. Being able to identify different histogram shapes and vocabulary associated with the shape.

2. Being able to calculate mean, median, and mode and identify potential pitfalls associated with each type.

Uniform Symmetric Skewed right or

positively skewed

Skewed left or

negatively skewed Uniform

Other Important information:

Sometimes a histogram has a single, central, or several different peaks. These peaks are called ____________________.

Therefore the mode is _____________________________________________

A histogram with:

1 mode: ___________________ 2 modes: __________________ 3+ Peaks: ____________________ no mode ___________________

What type of data might cause a bimodal distribution?

Central Tendency

Excerpt taken from Naked Statistics, by Charles Wheelan, regarding the economic health of the middle class:

From baseball to income, the most basic task when working with data is to summarize a great deal of

information. There are some 330 million residents in the United States. A spreadsheet with the name

and income history of every American would contain all the information we could ever want about the

economic health of the country-yet it would also be so unwieldy as to tell us nothing at all. The irony is

that more data can often present less clarity. So we simplify. We perform calculations that reduce a

complex array of data into a handful of numbers that describe those data, just as we might

encapsulate a complex, multifaceted Olympic gymnastics performance with one number: 9.8

Measures of central tendency are one way that we can simplify the data.

Example: You are currently taking 7 classes at Hinsdale Central and the following is a data set representing the

number of tests that you take per class in the 3rd quarter: {0, 1, 1, 2, 2, 2, 4}

What are the three measures of central tendency?

1.

2.

3.

How do you calculate each of them using the example above?

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Pitfalls:

Example: Four Microsoft entry level workers are at a lunch table. Their annual salaries are {$40,250, $42,000,

$37,800, $31,350}.

a. What is the mean of this data?

b. Bill Gates makes a surprise visit and sits at the lunch table with the new workers. Bill Gates’ annual

salary is $1 billion dollars. What would be the mean salary for these five employees?

The mean can be _________________________ by ___________________________.

Example: A company sells printers. The company did an analysis of customer complaints per printer. Two

frequency distributions were calculated...one for the company and one for the biggest competitor.

After some quick calculations, you determine your company has a median number of complaints of 1, and your

biggest competitor has a median number of complaints of 2. You conclude that your company is doing great and

your printers are performing well. Why is this not so?

The median can ____________________________ ___________________________________.

SAT Examples:

Example 1 Mr. Jones raised all his students’ math scores on a recent exam by 10 points. What affect did this have

on the mean and median of the scores?

A. The mean increased by 10 points but the median remained the same.

B. The median increased by 10 points, but the mean remained the same.

C. The mean increased by 10 points and the median increased by 10 points.

D. The mean and the median remained the same.

Example 2: Which of the following CANNOT change the value of the median in a set of five numbers?

A. Adding 0 to the set

B. Multiplying each value by -1

C. Increasing the least value only

D. Increasing the greatest value only

E. Squaring each value

Day 1 Homework:

1. Describe the distribution according to the shape of the histogram: symmetric or skewed.

Also, could the distribution possibly be uniform, unimodal, or bimodal?

a. b. c.

[[

d. e. f.

2. Would you expect the distributions of these variables to be symmetric or skewed?

Also, could the distribution possibly be uniform, unimodal, or bimodal? Explain why?

a. The number of speeding tickets each student in the senior class of a college class has ever had.

b. Weights of female babies born in a particular hospital over the course of a year

c. Number of times each face of a die shows in 100 tosses.

3. Which histogram does not belong with the other three? Explain your reasoning.

A. B.

4. The table show the number of tickets that are sold for various

Baseball games in a league over an entire season. Display the data

in a histogram. Describe the shape of the distribution.

5. Would data that is skewed left or skewed right explain more successful final exam scores? Why?

Explain with a histogram sketch.

6. The histogram display shows the heights of some of the singers in a coed chorus, collected so that the

singers could be positioned on stage with shorter ones in front and taller ones in back.

a. Describe the distribution

b. Can you account for the features you see? (Hint: 2 peaks)

7. You want to start playing on a club basketball team and want to join the best team. There are three

different basketball teams in your area and each has played five games. The following table represents

the amount of points that the team scored per game.

Team Game

1

Game

2

Game

3

Game

4

Game

5

Tigers 67 87 54 99 78

Wolves 85 90 44 80 46

Lions 32 101 65 88 55

a. If you rank each team by their mean scores, which team would you join?

b. Instead of using mean scores, you use the median score of each team to make your decision.

Which team do you join?

c. Pretend you are the coach of the Lions and you were being interviewed about your team for the

local newspaper. Would it be better for you to report your mean score or your median score?

8. The mode of the set below are 2 and 11. What is one possible value of x + y?

{2, 3, 9, 4, 11, 4x – 8, 3y – 4}

9.

The table above show the temperatures at noon, in degrees Fahrenheit, in a city in Hawaii over a one-week perod. If m

represents the median of these temperatures, f represents the temperature that occurred most often, and a reprsents

the average (arithmetic mean) of these seven temperatures, which of the following is the correct order of m, f, and a?

(A) a < m < f

(B) a < f < m

(C) m < a < f

(D) m < f < a

(E) a = m < f