Statistics and Data (Algebraic)

Sec. 9.7aSec. 9.7a

Statistics and Statistics and Data (Algebraic)Data (Algebraic)

Some Definitions…Some Definitions…

Statistic – numbers associated with a data set

(when used to describe the individuals in the data set,they are called descriptive statistics)

Parameter – numbers associated with an entire population

(we gather information from samples of thepopulation, then use inferential statistics to

make inferences about parameters)


A 1996 study reported that 33% of adolescents say there is noadult at home when they return from school. The report wasbased on a survey of 600 randomly selected people aged 12 to17 years old and had a margin of error of +4%. Did the surveymeasure a parameter or a statistic, and what does that “marginof error” mean?

The survey did not measure all adolescents in the population,so it did not measure a parameter. They sampled 600adolescents and found a statistic…

However, note that the first sentence is making an inferenceabout all American adolescents…


A 1996 study reported that 33% of adolescents say there is noadult at home when they return from school. The report wasbased on a survey of 600 randomly selected people aged 12 to17 years old and had a margin of error of +4%. Did the surveymeasure a parameter or a statistic, and what does that “marginof error” mean?

Interpret the margin of error as meaning “between 29% and37% of all American adolescents would say that there is noadult home when they return from school.”


1 2, , , nx x x

What is the mathematical meaning of the word “average?”

Three possible meanings, all of them measures of center.

The mean of a list of n numbers is

The mean is also called the arithmetic mean, arithmetic average,or average value.

1 2

1

1 nn

ii

x x xX x

n n

EX: “The average on last week’s test was 83.4.”


1 2, , , nx x x



The median of a list of n numbers

arranged in order (either ascending or descending) is

• the middle number if n is odd, and

EX: “The average test score puts you right in the middleof the class.”

• the mean of the two middle numbers if n is even.




The mode of a list of numbers is the number that appears mostfrequently in the list.

EX: “The average American student starts college at age 18.”

Note: A statistic is called resistant if it is not strongly affectedby outliers…………………………which of our three averageswould be considered resistant?

Guided PracticeGuided Practice

14,28,16,39,61,33,23,26,8,13,9,5

Find the mean, median, and mode of the annual home run totalsfor Roger Maris’s major league career:

Mean:14 28 9 5

12X

Is this statistic Is this statistic resistant resistant ??

27522.9

12

Not really…Not really…


14,28,16,39,61,33,23,26,8,13,9,5


Median:16 23

2

Is this statistic Is this statistic resistant resistant ??

19.5Much more so thanMuch more so than the mean…the mean…

To find the median, first write the data set in order:

5,8,9,13,14,16,23,26,28,33,39,61Because there are 12 numbers, we average the middle two:


14,28,16,39,61,33,23,26,8,13,9,5


This data set has no mode!!!

The mode is typically the least important measureThe mode is typically the least important measureof center, but it sometimes has statistical significance…of center, but it sometimes has statistical significance…

How about the mode???

So what’s the mode for Hank Aaron’s home run totals?(see Table 9.8 on p.764)

Mode: 44

Guided PracticeGuided PracticeA teacher gives a 10-point quiz and records the scores in afrequency table shown below. Find the mode, median, andmean of the data set.

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

Frequency 2 2 3 8 4 3 3 2 1 1 1

First, how many total scores are there?

Add the frequencies Add the frequencies there are 30 scores there are 30 scores

To find the mode, look for the score with the highest frequency.

Mode: 7Mode: 7

Guided PracticeGuided PracticeA teacher gives a 10-point quiz and records the scores in afrequency table shown below. Find the mode, median, andmean of the data set.

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

Frequency 2 2 3 8 4 3 3 2 1 1 1

The median will be the mean of the 15th and 16th numbers.

Median: 6.5Median: 6.5

Count the frequencies from left to right until we come to 15.

The 15th number is 7, and the 16th number is 6.


10 2 9 2 8 3 7 8 6 4 5 3

4 3 3 2 2 1 1 1 0 1

30

A teacher gives a 10-point quiz and records the scores in afrequency table shown below. Find the mode, median, andmean of the data set.

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

Frequency 2 2 3 8 4 3 3 2 1 1 1

To find the mean, multiply each number by its frequency, add theproducts, and divide the total by 30:

Mean: 5.93Mean: 5.93


88.5 0.75 0.25 89.5x

Let’s try a problem that uses the concept of weighted mean:

At a certain school, it is a policy that the final exam must count25% of the final grade. If Sam has an 88.5 average going intothe final exam, what is the minimum exam score needed toearn a 90 for the semester?

Sam needs to make at least a 92.5 on the final exam.Sam needs to make at least a 92.5 on the final exam.

Assume that an 89.5 will be rounded up to a 90 on the transcript:

92.5x

The Five-The Five-Number Number

SummarySummary

The Five-Number The Five-Number SummarySummary

The measures of center from last class tell part of the story, butwe also need measures of spread.

Range – the difference between the maximum and minimumvalues in a data set.

Quartiles – separate a data set into fourths (just as the medianseparates a data set into halves)

First Quartile (Q ) – the median of the lower half of the data

Second Quartile – the median

Third Quartile (Q ) – the median of the upper half of the data

1

3

The Five-Number The Five-Number SummarySummary

The measures of center from last class tell part of the story, butwe also need measures of spread.

Interquartile Range (IQR) – measures the spread between thefirst and third quartiles (comprises the middle half of the data):

IQR = Q – Q

Definition: Five-Number SummaryDefinition: Five-Number Summary

The five-number summary of a data set is the collection:

{minimum, Q , median, Q , maximum}

13

31

Find the five-number summary for the male and female lifeexpectancies in South American nations (Table 9.12 on p.768)and compare the spreads.


Males:{59.0, 60.5, 61.5, 66.7, 67.9, 68.5, 69.0, 70.3, 71.4, 71.9, 72.1, 72.6}

Females:{66.2, 66.7, 67.7, 72.8, 74.3, 74.4, 74.6, 76.5, 76.6, 78.8, 79.0, 79.4}

Five-Number Summaries:

Males:

59.0,64.1,68.75,71.65,72.6

Females: 66.2,70.25,74.5,77.7,79.4

Range: 72.6 – 59.0 = 13.6, IQR = 71.65 – 64.1 = 7.55

Range: 79.4 – 66.2 = 13.2, IQR = 77.7 – 70.25 = 7.45

Guided PracticeGuided PracticeFive-Number Summaries:

Males: 59.0,64.1,68.75,71.65,72.6

Females: 66.2,70.25,74.5,77.7,79.4

Range: 72.6 – 59.0 = 13.6, IQR = 71.65 – 64.1 = 7.55

Range: 79.4 – 66.2 = 13.2, IQR = 77.7 – 70.25 = 7.45

Not only do the women live longer, but there is lessvariability in their life expectancies (as measured by IQR).Male life expectancy is more strongly affected by differentpolitical conditions within countries (war, crime, etc.).

The The shapes shapes of of distributionsdistributions

Of the two histograms shown below, which displays a data setwith more variability? Explain your answer.

The extreme values in (a) cause the range to be big, but thecompact distribution indicate a small IQR. The data in (b)exhibit high variability.

(a) (b)

The The shapes shapes of of distributionsdistributions

Compare the medians and means for the data displayed in thethree histograms below.

SymmetricDistribution

(a) (b) (c)

Skewed RightDistribution

Skewed LeftDistribution

Mean = Median Mean > Median Mean < Median

Determine the five-number summary, the range, and the IQR forthe annual home run production data for Mark McGwire andBarry Bonds (Table 9.6 on p.763).


McGwire{ 3, 9, 9, 22, 29, 32, 32, 33, 39, 39, 42, 49, 52, 58, 65, 70 }

Note: The underlined numbers are those of interest for thefive-number summary.

Five-Number Summary: { 3, 25.5, 36, 50.5, 70 }

Range: 70 – 3 = 67

IQR: 50.5 – 25.5 = 25

No Outliers

Determine the five-number summary, the range, and the IQR forthe annual home run production data for Mark McGwire andBarry Bonds (Table 9.6 on p.763).


Bonds{ 16, 19, 24, 25, 25, 33, 33, 34, 34, 37, 37, 40, 42, 46, 49, 73 }

Note: The underlined numbers are those of interest for thefive-number summary.

Five-Number Summary: { 16, 25, 34, 41, 73 }

Range: 73 – 16 = 57

IQR: 41 – 25 = 16

Outlier: 73

Statistics and Data (Algebraic)

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