1.21.2NN

Displaying Quantitative Displaying Quantitative Data with GraphsData with Graphs

(dot plots, stemplots, histograms, shape)(dot plots, stemplots, histograms, shape)

Target: Target: I can graph quantitative data using I can graph quantitative data using dotplots and stemplots. dotplots and stemplots.

I can describe distributions using I can describe distributions using shape, center, spread and outliers. shape, center, spread and outliers.

h.w: pg 42: 37, 40, 41, 44, 45h.w: pg 42: 37, 40, 41, 44, 45

Displaying Quantitative Displaying Quantitative VariablesVariables

Quantitative VariablesQuantitative Variables: take numerical : take numerical value for which it makes sense to find value for which it makes sense to find an average.an average.

Dotplots and StemplotsDotplots and Stemplots• Dotplots are a good way to display Dotplots are a good way to display small data small data

sets. sets.

• Stemplots are better for Stemplots are better for larger data sets larger data sets (medium to large).(medium to large).

Dotplots Dotplots OOne of simplest types of graphs to ne of simplest types of graphs to

display quantitative datadisplay quantitative data

Remember: Remember:

•label axis label axis •title graph title graph •scale axisscale axis

Stem (and leaf) plotStem (and leaf) plot (directions (directions p. 33)p. 33)

Graphs quantitative data and is used for Graphs quantitative data and is used for small small or medium sized data.or medium sized data. (Very common graph). (Very common graph).

Ex. 3 Watch the Caffeine Ex. 3 Watch the Caffeine (See data on p.13, don’t (See data on p.13, don’t enter)enter)

• Tip for organizing the data: First enter data Tip for organizing the data: First enter data into list. Then sort.into list. Then sort.

How to construct a How to construct a stemplotstemplot

1. Separate each observation into a stem consisting of 1. Separate each observation into a stem consisting of all all but the but the rightmost digitrightmost digit (1 digit leafs). (1 digit leafs).

2. Write stems vertically in increasing order and draw a 2. Write stems vertically in increasing order and draw a vertical line to their right; write each leaf to right of its vertical line to their right; write each leaf to right of its stem.stem.

3. If data not entered into calculator list, write stems again 3. If data not entered into calculator list, write stems again and rearrange leaves in increasing order out from stem and rearrange leaves in increasing order out from stem (don’t need to do if you sort with calculator).(don’t need to do if you sort with calculator).

How to construct a stemplot How to construct a stemplot cont.cont.

4. 4. Title graphTitle graph and add and add keykey describing describing what stems and leaves represent what stems and leaves represent Key: 3/5 means the soft drink contains 35 mg of caffeine per 8 ounce serving.

Split-stem and leaf plotSplit-stem and leaf plot

• When splitting stems, be sure each stem is When splitting stems, be sure each stem is assigned an equal number of possible leaf digits.assigned an equal number of possible leaf digits.

• Notice graph “a” is very “skyscraper-ish”. Notice graph “a” is very “skyscraper-ish”.

• So split-stem.So split-stem.

Back-to-back stemplotsBack-to-back stemplots - - good to compare two sets of good to compare two sets of data data

Remember: 5 stems is a good minimum Remember: 5 stems is a good minimum • AdvantagesAdvantages:: easy to construct; display actual easy to construct; display actual

data values data values • DisadvantagesDisadvantages:: does not work well with large does not work well with large

data setsdata sets

Example: Who’s Example: Who’s Taller?Taller?

• Which gender is taller, males or females? Which gender is taller, males or females? A A sample of 14-year-olds from the United sample of 14-year-olds from the United Kingdom was randomly selected using the Kingdom was randomly selected using the Census At SchoolCensus At School website. website.

• Here are the heights of the students (in cm):Here are the heights of the students (in cm):

• Construct stemplots. Construct stemplots. Report back.Report back.

• Male: 154, 157, 187, 163, 167, 159, 169, 162, Male: 154, 157, 187, 163, 167, 159, 169, 162, 176, 177, 151, 175, 174, 165, 165, 183, 180176, 177, 151, 175, 174, 165, 165, 183, 180

• Female: 160, 169, 152, 167, 164, 163, 160, Female: 160, 169, 152, 167, 164, 163, 160, 163, 169, 157, 158, 153, 161, 165, 165, 159, 163, 169, 157, 158, 153, 161, 165, 165, 159, 168, 153, 166, 158, 158, 166168, 153, 166, 158, 158, 166

Describing the Overall Pattern of a Describing the Overall Pattern of a DistributionDistribution

• See if you can describe the See if you can describe the shapeshape of of the dist. in a few words. the dist. in a few words. (long tail to the (long tail to the right/left, skewed, unimodal, bimodal-right/left, skewed, unimodal, bimodal-2peaked)2peaked)

• Identify Identify outliersoutliers: : indiv. observations indiv. observations that that fall outside the overall pattern fall outside the overall pattern of of the graph.the graph.

• Give Give centercenter and and spreadspread..

Don’t forget your SOCS!

Don’t forget your SOCS!

Patterns of DistributionsPatterns of Distributions

CenterCenter - A value that divides the - A value that divides the observations so that about observations so that about half half take take larger values and about half have larger values and about half have smaller values.smaller values.

Median (M) Median (M) -- The The midpointmidpoint of a of a distribution, the # such that half the distribution, the # such that half the observations are smaller and the observations are smaller and the other half are larger.other half are larger.

To find the median of a To find the median of a distribution:distribution:

1. Arrange all observations from 1. Arrange all observations from smallest to largest. smallest to largest.

2. If n is odd, M is the center 2. If n is odd, M is the center observation in the ordered list.observation in the ordered list.

2, 3, 42, 3, 4

2, 3, 3, 4, 5 2, 3, 3, 4, 5

(n + 1)/2 (n + 1)/2 thth observation is the observation is the medianmedian

3. If n is even, M (median) is the 3. If n is even, M (median) is the mean mean of the two centerof the two center observations in the observations in the ordered list.ordered list.

1, 2, 3, 5;1, 2, 3, 5; 10, 15, 21, 23, 10, 15, 21, 23, 30, 40 30, 40

1, 1, 2, 3,2, 3, 5; 10, 15, 5; 10, 15, 21, 2321, 23, 30, 40, 30, 40M = 2.5M = 2.5 M = 22M = 22

Which observations if list is long?Which observations if list is long?Take mean of the Take mean of the (n/2) (n/2) thth observation observation and the next observationand the next observation..

We will usually find median with the We will usually find median with the calculator. calculator.

• Enter data into L1; STAT:EDITEnter data into L1; STAT:EDIT

• Highlight L1 and press clear if Highlight L1 and press clear if needed.needed.

• Enter data. Ex. 1,1,2,4,6,7,9,10Enter data. Ex. 1,1,2,4,6,7,9,10

• 22ndnd STAT: MATH: median(L1): STAT: MATH: median(L1): enter enter

SpreadSpread - Can be described by - Can be described by giving giving smallest and largest valuessmallest and largest values..

ShapeShape - Shape of graph of - Shape of graph of distribution.distribution.

SymmetricSymmetric - Right and left sides of histogram - Right and left sides of histogram are approximate mirror images of each other.are approximate mirror images of each other.

SkewedSkewed to the right to the right (positively skewed(positively skewed))

• Right side of histogram Right side of histogram extends much farther extends much farther out out than left sidethan left side

• pulled to the rightpulled to the right

• long tail to the right long tail to the right

Skewed to the left Skewed to the left (negatively skewed)(negatively skewed)

Left side of histogram extends much Left side of histogram extends much farther out than right side; pulled to farther out than right side; pulled to the left, long tail to the left. (Draw)the left, long tail to the left. (Draw)

Ex 4.Ex 4. Michigan College Michigan College Tuitions Tuitions

There are There are 81 colleges81 colleges and and universities in Michigan. universities in Michigan.

Their tuition and fees for the 1999 to Their tuition and fees for the 1999 to 2000 school year run from 2000 school year run from $1260 at $1260 at Kalamazoo Valley Community Kalamazoo Valley Community College to $19,258 at Kalamazoo College to $19,258 at Kalamazoo CollegeCollege. .

Complete and report back (4 min).Complete and report back (4 min).

Ex 4. Michigan College Tuitions Ex 4. Michigan College Tuitions cont.cont.a.a. What do the What do the stemsstems

and and leavesleaves represent in the represent in the stemplot? ($1260 to stemplot? ($1260 to $19,258 )$19,258 )

Stems = thousandsStems = thousands

Leaves = hundredsLeaves = hundreds

Have the data been Have the data been rounded? rounded?

The data have been The data have been rounded to the rounded to the nearest $100.nearest $100.

Note: stems can be multiple digits, leaves can only be the right most digit.

b. Describe the shape, center, and b. Describe the shape, center, and spread.spread.

Shape:Shape:

The distribution isThe distribution is skewed strongly to skewed strongly to the right the right with a peak with a peak at the one stem.at the one stem.

Center:Center:

45 (approx. $4500)45 (approx. $4500)

Spread:Spread:

approx. 19,000 approx. 19,000 ($1300 to $19300)($1300 to $19300)

Median is 41st value – count them.

Are there any outliers?Are there any outliers?

• The observations The observations 182 and 193 182 and 193 appear to be appear to be outliers. ($18,200, outliers. ($18,200, $19,300)$19,300)

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Comparing DistributionsComparing Distributions

Some of the most interesting statistics questions Some of the most interesting statistics questions involve comparing two or more groups.involve comparing two or more groups.

– Always discuss Always discuss shape, center, spread, and possible shape, center, spread, and possible outliersoutliers whenever you compare distributions of a whenever you compare distributions of a quantitative variable.quantitative variable.

Household_Size0 5 10 15 20 25 30

Household_Size0 5 10 15 20 25 30

Compare the distributions of household size for these two countries. Don’t forget your SOCS!Don’t discussing SOCS for each distribution separately. You will not receive credit!Use phrases like “about the same as” or “is much greater than” to receive full credit.

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Think – Pair - Think – Pair - ShareShare

• In small group, read over In small group, read over comparing distributions comparing distributions example on page 32.example on page 32.

• Do problem #43 on page 43.Do problem #43 on page 43.

• Report out.Report out.