Slide 4 - 1Copyright © 2010 Pearson Education, Inc.

Suppose that 30% of the subscribers to a cable television service watch the shopping channel at least once a week. You are to design a simulation to estimate the probability that none of five randomly selected subscribers watches the shopping channel at least once a week. Which of the following assignments of the digits 0 through 9 would be appropriate for modeling an individual subscriber’s behavior in this simulation?

a.a. Assign “0,1,2” as watching the shopping channel at least Assign “0,1,2” as watching the shopping channel at least once a week and “3,4,5,6,7,8, and 9” as not watching.once a week and “3,4,5,6,7,8, and 9” as not watching.

b.b. Assign “0,1,2,3” as watching the shopping channel at least Assign “0,1,2,3” as watching the shopping channel at least once a week and “4,5,6,7,8, and 9” as not watching.once a week and “4,5,6,7,8, and 9” as not watching.

c.c. Assign “0,1,2,3,4” as watching the shopping channel at Assign “0,1,2,3,4” as watching the shopping channel at least once a week and “5,6,7,8, and 9” as not watching.least once a week and “5,6,7,8, and 9” as not watching.

d.d. Assign “0” as watching the shopping channel at least once Assign “0” as watching the shopping channel at least once a week and “1,2,3,4 and 5” as not watching; ignore digits a week and “1,2,3,4 and 5” as not watching; ignore digits “6,7,8, and 9”“6,7,8, and 9”

e.e. Assign “3” as watching the shopping channel at least once Assign “3” as watching the shopping channel at least once a week and “1,2,3,4,5,6,7,8 and 9” as not watching.a week and “1,2,3,4,5,6,7,8 and 9” as not watching.

Slide 4 - 2Copyright © 2010 Pearson Education, Inc.

Suppose that 30% of the subscribers to a cable television service watch the shopping channel at least once a week. You are to design a simulation to estimate the probability that none of five randomly selected subscribers watches the shopping channel at least once a week. Which of the following assignments of the digits 0 through 9 would be appropriate for modeling an individual subscriber’s behavior in this simulation?

a.a. Assign “0,1,2” as watching the shopping channel at least Assign “0,1,2” as watching the shopping channel at least once a week and “3,4,5,6,7,8, and 9” as not watching.once a week and “3,4,5,6,7,8, and 9” as not watching.

b.b. Assign “0,1,2,3” as watching the shopping channel at least Assign “0,1,2,3” as watching the shopping channel at least once a week and “4,5,6,7,8, and 9” as not watching.once a week and “4,5,6,7,8, and 9” as not watching.

c.c. Assign “0,1,2,3,4” as watching the shopping channel at Assign “0,1,2,3,4” as watching the shopping channel at least once a week and “5,6,7,8, and 9” as not watching.least once a week and “5,6,7,8, and 9” as not watching.

d.d. Assign “0” as watching the shopping channel at least once Assign “0” as watching the shopping channel at least once a week and “1,2,3,4 and 5” as not watching; ignore digits a week and “1,2,3,4 and 5” as not watching; ignore digits “6,7,8, and 9”“6,7,8, and 9”

e.e. Assign “3” as watching the shopping channel at least once Assign “3” as watching the shopping channel at least once a week and “1,2,3,4,5,6,7,8 and 9” as not watching.a week and “1,2,3,4,5,6,7,8 and 9” as not watching.

Copyright © 2010 Pearson Education, Inc.

Chapter 4Displaying and Summarizing Quantitative Data

Slide 4 - 4Copyright © 2010 Pearson Education, Inc.

What type of data is used in bar charts or pie charts?

Histograms – Displays quantitative variables by using bars of equal widths with counts on the vertical axis. They do not show data values.

Here is a histogram of earthquake magnitudes:

Slide 4 - 5Copyright © 2010 Pearson Education, Inc.

A relative frequency histogram displays the percentage of cases in each bin instead of the count.

Spaces are gaps in the data.

Here is a relative frequency histogram of earthquake magnitudes:

Slide 4 - 6Copyright © 2010 Pearson Education, Inc.

TI Tips – Making a Histogram Enter the following scores in L1: 22, 17,18, 29,

22, 22, 23, 24, 23, 17, 21, 25, 20, 12, 19, 28, 24, 22, 21, 25, 26, 25, 16, 27, 22

2nd STATPLOT, choose Plot1, Enter In the Plot1 screen choose On, select the

histogram icon, then specify Xlist:L1 and Freq: 1 Turn off any other graphs and check y= for any

equations. ZOOM, select 9:ZoomStat, Enter Not bad, but the scale is a little messy. Reset

the scale to sensible values. WINDOW, Xmin = ?, Xmax=? Xscl=? GRAPH (if we hit ZoomStat it will undo our

scale) TRACE

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TI Tips – Making a Histogram

Suppose you are given a set of test scores, with two grades in the 60s, four in the 70s, seven in the 80s, five in the 90s and one 100.

Enter the group cutoffs 60, 70, 80, 90, 100 in L2. Enter the corresponding frequencies 2,4, 7, 5, 1 in

L3. STATPLOT, Xlist:L2, Freq: L3 How would you set up the window?

Slide 4 - 8Copyright © 2010 Pearson Education, Inc.

Stem-and-Leaf Displays

Stem-and-leaf displays show the distribution of a quantitative variable, like histograms do, while preserving the individual values.

Stem-and-leaf displays contain all the information found in a histogram and, when carefully drawn, satisfy the area principle and show the distribution.

Slide 4 - 9Copyright © 2010 Pearson Education, Inc.

Stem-and-Leaf Example

Compare the histogram and stem-and-leaf display for the pulse rates of 24 women at a health clinic. Which graphical display do you prefer?

Slide 4 - 10Copyright © 2010 Pearson Education, Inc.

Dotplots

A dotplot is a simple display. It just places a dot along an axis for each case in the data.

The dotplot to the right shows Kentucky Derby winning times, plotting each race as its own dot.

Dotplots can be displayed horizontally or vertically.

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Think Before You Draw

Before making a stem-and-leaf display, a histogram, or a dotplot, check the Quantitative Data Condition: The data are

values of a quantitative variable whose units are known.

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Shape, Center, and Spread

When describing a distribution, make sure to always tell about three things: shape, center, and spread.

Slide 4 - 13Copyright © 2010 Pearson Education, Inc.

What is the Shape of the Distribution?

1. Does the histogram have a single, central hump or several separated humps?

2. Is the histogram symmetric?

3. Do any unusual features stick out?

Slide 4 - 14Copyright © 2010 Pearson Education, Inc.

Humps

1. Does the histogram have a single, central hump or several separated bumps?

Humps in a histogram are called modes. A histogram with one main peak is dubbed

unimodal; histograms with two peaks are bimodal; histograms with three or more peaks are called multimodal.

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Humps (cont.)

A bimodal histogram has two apparent peaks:

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Humps (cont.)

A histogram that doesn’t appear to have any mode and in which all the bars are approximately the same height is called uniform:

Slide 4 - 17Copyright © 2010 Pearson Education, Inc.

Symmetry

2. Is the histogram symmetric? If you can fold the histogram along a vertical line

through the middle and have the edges match pretty closely, the histogram is symmetric.

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Symmetry (cont.) The (usually) thinner ends of a distribution are called

the tails. If one tail stretches out farther than the other, the histogram is said to be skewed to the side of the longer tail.

In the figure below, the histogram on the left is said to be skewed left, while the histogram on the right is said to be skewed right.

Slide 4 - 19Copyright © 2010 Pearson Education, Inc.

Anything Unusual?

3. Do any unusual features stick out? Sometimes it’s the unusual features that tell

us something interesting or exciting about the data.

You should always mention any stragglers, or outliers, that stand off away from the body of the distribution.

Are there any gaps in the distribution? If so, we might have data from more than one group.

Slide 4 - 20Copyright © 2010 Pearson Education, Inc.

Anything Unusual? (cont.)

The following histogram has outliers—there are three cities in the leftmost bar:

Slide 4 - 21Copyright © 2010 Pearson Education, Inc.

Where is the Center of the Distribution?

If you had to pick a single number to describe all the data what would you pick?

It’s easy to find the center when a histogram is unimodal and symmetric—it’s right in the middle.

On the other hand, it’s not so easy to find the center of a skewed histogram or a histogram with more than one mode.

Slide 4 - 22Copyright © 2010 Pearson Education, Inc.

Center of a Distribution -- Median

The median is the value with exactly half the data values below it and half above it. It is the middle data

value (once the data values have been ordered) that divides the histogram into two equal areas.

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Spread

Are the values of the distribution tightly clustered around the center or more spread out?

The range of the data is the difference between the maximum and minimum values:

Range = max – min A disadvantage of the range is that a single

extreme value can make it very large and, thus, not representative of the data overall.

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Spread: The Interquartile Range

The interquartile range (IQR) lets us ignore extreme data values and concentrate on the middle of the data.

To find the IQR, we first need to know what quartiles are…

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Spread: The Interquartile Range (cont.)

Quartiles divide the data into four equal sections. One quarter of the data lies below the lower

quartile, Q1 One quarter of the data lies above the upper

quartile, Q3. The quartiles border the middle half of the data.

The difference between the quartiles is the interquartile range (IQR), so

IQR = upper quartile – lower quartile

Slide 4 - 26Copyright © 2010 Pearson Education, Inc.

Example: Find Range, Q1, Q2 (median), Q3 and the IQR of 1, 1, 2, 3, 4, 5, 6

Example: Find Range, Q1, Q2 (median), Q3 and the IQR of 1, 1, 2, 3, 4, 5, 6, 7

Slide 4 - 27Copyright © 2010 Pearson Education, Inc.

Spread: The Interquartile Range (cont.)

The lower and upper quartiles are the 25th and 75th percentiles of the data, so…

The IQR contains the middle 50% of the values of the distribution, as shown in figure:

Slide 4 - 28Copyright © 2010 Pearson Education, Inc.

5-Number Summary

The 5-number summary of a distribution reports its median, quartiles, and extremes (maximum and minimum)

The 5-number summary for the recent tsunami earthquake Magnitudes looks like this:

Slide 4 - 29Copyright © 2010 Pearson Education, Inc.

Center – Symmetric Distributions

When we have symmetric data, there is an alternative other than the median … the mean!

We use the Greek letter sigma to mean “sum” and write:

The formula says that to find the mean, we add up all the values of the variable and divide by the number of data values, n.

y Total

n

yn

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Summarizing Symmetric Distributions -- The Mean (cont.)

The mean feels like the center because it is the point where the histogram balances:

Slide 4 - 31Copyright © 2010 Pearson Education, Inc.

Example: Draw three graphs (skewed left, right and symmetric) and roughly estimate where the mean and median will be located in each.

Slide 4 - 32Copyright © 2010 Pearson Education, Inc.

What About Spread? The Standard Deviation

A more powerful measure of spread than the IQR is the standard deviation, which takes into account how far each data value is from the mean.

A deviation is the distance that a data value is from the mean.

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What About Spread? The Standard Deviation (cont.)

The variance, notated by s2, is found by summing the squared deviations and (almost) averaging them:

The variance will play a role later in our study, but it is problematic as a measure of spread—it is measured in squared units!

s2 y y 2n 1

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What About Spread? The Standard Deviation (cont.)

The standard deviation, s, is just the square root of the variance and is measured in the same units as the original data.

s y y 2n 1

Slide 4 - 35Copyright © 2010 Pearson Education, Inc.

Thinking About Variation

When the data values are tightly clustered around the center of the distribution, the IQR and standard deviation will be small.

When the data values are scattered far from the center, the IQR and standard deviation will be large.

Slide 4 - 36Copyright © 2010 Pearson Education, Inc.

Tell -- What About Unusual Features?

If there are multiple modes, try to understand why. If you identify a reason for the separate modes, it may be good to split the data into two groups.

If there are any clear outliers and you are reporting the mean and standard deviation, report them with the outliers present and with the outliers removed. The differences may be quite revealing. Note: The median and IQR are not likely to be

affected by the outliers.

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What Can Go Wrong?

Don’t make a histogram of a categorical variable—bar charts or pie charts should be used for categorical data.

Don’t look for shape, center, and spread of a bar chart.

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What Can Go Wrong? (cont.)

Choose a bin width appropriate to the data. Changing the bin width changes the

appearance of the histogram:

Slide 4 - 39Copyright © 2010 Pearson Education, Inc.

TI- Tips Calculating Numerical Summaries

Enter the following scores in L1: 22, 17,18, 29, 22, 22, 23, 24, 23, 17, 21, 25, 20, 12, 19, 28, 24, 22, 21, 25, 26, 25, 16, 27, 22

Go to STAT, CALC, select 1-Var Stats, Enter

Enter again. (If your data weren’t in L1, you would have to specify where it was before hitting enter.)

Cursor down! Sx is the standard

deviation

Slide 4 - 40Copyright © 2010 Pearson Education, Inc.

TI Tips –

Suppose you are given a set of test scores, with two grades in the 60s, four in the 70s, seven in the 80s, five in the 90s and one 100.

Enter the group cutoffs 60, 70, 80, 90, 100 in L2. Enter the corresponding frequencies 2, 4, 7, 5, 1 in

L3. 1-Var Stats L2, L3 (1-Var Stats Value, Frequencies)

Slide 4 - 41Copyright © 2010 Pearson Education, Inc.

Homework Pg. 72 5 – 45 (15 don’t do by hand use calculator, skip 37, 41)