Displaying Distributions with Displaying Distributions with GraphsGraphs

2.1a 2.1a

h.w: pg 105: 2, 5, 10, 11, 13 - 15 h.w: pg 105: 2, 5, 10, 11, 13 - 15

Target Goal: I can construct and interpret an ogive Target Goal: I can construct and interpret an ogive (relative cumulative frequency graph).(relative cumulative frequency graph).

Relative frequency, cumulative Relative frequency, cumulative frequency, percentiles, and ogives.frequency, percentiles, and ogives.

Percentile Percentile (pth percentile)(pth percentile) - The value such that - The value such that “p” percent of the observations fall “p” percent of the observations fall at or below it.at or below it.

Relative frequencyRelative frequency - - Divide count in each class Divide count in each class by total numberby total number then multiply by 100 to convert then multiply by 100 to convert to a percentage; “percent” . . . think: getting to a percentage; “percent” . . . think: getting ready to do a pie chart.ready to do a pie chart.

Cumulative frequencyCumulative frequency - Add counts in - Add counts in frequency column that fall frequency column that fall in or below current in or below current class interval. class interval.

Relative Cumulative FrequencyRelative Cumulative Frequency

Divide entries in cumulative frequency Divide entries in cumulative frequency column by total number. column by total number. OgiveOgive pronounced pronounced [O-JIVE]. [O-JIVE]. (presidents age)(presidents age)

(Percents)(Percents) (Ogive)(Ogive) (Counts)(Counts) Relative Relative Cum. Relative Cum. Relative

ClassClass Frequency Freq. Freq. Cum. Freq. Frequency Freq. Freq. Cum. Freq.40-4440-44 2 2 2/44 = 4.5% 2 2/44 = 4.5% 2/44 = 4.5% 2 2/44 = 4.5%45-4945-49 7 7 7/44 = 15.9% 9 7/44 = 15.9% 9 9/44 = 20.5% 9/44 = 20.5%50-5450-54 13 1355-5955-59 12 1260-6460-64 7 765-6965-69 3 3TotalTotal 44 44

OGIVE: Relative cumulative OGIVE: Relative cumulative frequency graphfrequency graph

Graph of relative standing of an individual Graph of relative standing of an individual observation. observation.

Various info you can find.Various info you can find.

6060thth % tile occurs at % tile occurs at age 55 is at about theage 55 is at about theBill Clinton Bill Clinton = 46 years= 46 years:: About of all U.S. presidents were the same age as or younger About of all U.S. presidents were the same age as or younger

than Bill Clinton orthan Bill Clinton or Bill Clinton was younger than of all U.S. presidents when Bill Clinton was younger than of all U.S. presidents when

inaugurated. inaugurated.

57age50th percentile

10%

90%about

Pg. 88How old was Obama when he was elected?

Pg. 88

Example: Drive Time Example: Drive Time Professor Moore, who lives a few miles Professor Moore, who lives a few miles outside a college town, records the time outside a college town, records the time he takes to drive to the college each he takes to drive to the college each morning. Here are the times (in minutes) morning. Here are the times (in minutes) for 42 consecutive weekdays, with the for 42 consecutive weekdays, with the dates in order along the rows:dates in order along the rows:

d. Construct an Ogive for Professor Moore’s Drivetimes.d. Construct an Ogive for Professor Moore’s Drivetimes.Fill in first 2 rows of relative cum freq. table. We will fill in the Fill in first 2 rows of relative cum freq. table. We will fill in the rest of the graph together.rest of the graph together.Note: right most value in range corres. to relative cum. freq. % Note: right most value in range corres. to relative cum. freq. % on ogive.on ogive. Relative Cumulative RelativeRelative Cumulative Relative

Drivetime Frequency Frequency Frequency Cum.FreqDrivetime Frequency Frequency Frequency Cum.Freq..6.5-6.5-6.96.9 11 2.4%2.4% 11 2.4%2.4%7.0-7.0-7.47.4 22 4.8%4.8% 33 7.1%7.1%7.5-7.5-7.97.9 88 19.0%19.0% 1111 26.2%26.2%8.0-8.0-8.48.4 1111 26.2%26.2% 2222 52.4%52.4%8.5-8.5-9.09.0 1212 28.6%28.6% 3434 81.0%81.0%9.1-9.1-9.49.4 66 14.3%14.3% 4040 95.2%95.2%9.5-9.5-9.99.9 11 2.4%2.4% 4141 97.6%97.6%10.0-10.0-10.410.4 11 2.4%2.4% 4242 100%100%

Remember: begin ogive with a point at height 0% Remember: begin ogive with a point at height 0% at left endpoint of lowest class interval; last point at left endpoint of lowest class interval; last point

plotted should be at a point of 100%plotted should be at a point of 100%

60th percentile occurs at about age 5760th percentile occurs at about age 57

Graph Ogive (4 min)Graph Ogive (4 min)

Rel

ativ

e C

umul

ativ

e F

requ

ency

Drive Time (minutes)

7.57.0 8.0 8.5 9.0 9.5 10.0 10.50

102030405060

708090

100

6.5

Example: Drive Time cont.Example: Drive Time cont. e. Use your ogive e. Use your ogive (graph)(graph) from b. to estimate the from b. to estimate the

center and 90th percentile for the distribution. center and 90th percentile for the distribution. center:center: about 8.4 about 8.4

How did you find the center?How did you find the center? From graphFrom graph Or Or 2ndSTAT:MATH:median2ndSTAT:MATH:median By hand:By hand: 42 observations, the average of 42 observations, the average of

the the 2121stst and 22 and 22nd nd observation. (see table) observation. (see table) 9090thth percentile: percentile: approximately 9.4 approximately 9.4

f. Use your ogive to estimate the percentile corresponding f. Use your ogive to estimate the percentile corresponding to a drive time of 8.00 minutes.to a drive time of 8.00 minutes.

Rel

ativ

e C

umul

ativ

e F

requ

ency

Drive Time (minutes)

7.57.0 8.0 8.5 9.0 9.5 10.0 10.50

102030405060708090100

8.0 is approximately the 26th percentile8.0 is approximately the 26th percentile

The End The End