Interpreting Data

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Curriculum Ready

Interpreting Data

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Interpreting Data

The median of a data set is the

The median of a data set is the middle score. Does this mean that the number of middle score. Does this mean that the number of scores greaterscores greater than the median is the same as

than the median is the same as the number of scores less than the median?the number of scores less than the median?

If the median splits the data in

If the median splits the data in two halves, what do you think "quarles" do?two halves, what do you think "quarles" do?

The "range" of data is the

The "range" of data is the dierence between the highest and lowest score. What is dierence between the highest and lowest score. What is "interquar"interquarle range"?le range"?

INTERPRETING

DA

TA

Answer these quesons,

Answer these quesons, beforebefore working through the chapter. working through the chapter.

Answer these quesons,

Answer these quesons, after after working through the chapter. working through the chapter. Dieren

Dierent lists of t lists of data have dierendata have dierent properes. This unit t properes. This unit is focused on the is focused on the results and conclusions thatresults and conclusions that

can be found from these dierent properes.

But now I think: But now I think:

What do I know now that I didn’t know before?

I used to think: I used to think:

The median of a data set is the

The median of a data set is the middle score. Does this mean that the number of middle score. Does this mean that the number of scores greater than thescores greater than the median is the same as the number

median is the same as the number of scores less than the median?of scores less than the median?

If the median splits the data in

If the median splits the data in two halves, what do you think "quarles" do?two halves, what do you think "quarles" do?

The "range" of data is the

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Interpreting Data

Basics

a b c d e

Basic Statistics

Data is just a list of numbers called 'scores' or 'results'. The basic stascs that can be found from these scores are the mean, median or mode. (These are also called "measures of central tendency")

Also remember that the symbol

/

(called "sigma") means 'sum of' and so when

/

x is wrien, it means the 'sum of the scores'.

How many people had a height of 112cm?

The frequency of 112 is5. This means there were 5 people who are 112 cm tall.

The cumulave frequency of 113 is the sum of the frequencies for scores of 113 or less. The cf of113 is3 + 5 + 10 = 18

Find the mean height, x

r

.

Find the mode.

Find the median.

The mode is113 since it has the highest frequency (10).

The median is the score in the middle. Since there are 35 scores the middle posion is the18th _score.

What is the cumulave frequency of113 cm? Here is an example.

• The mean is the average score. The symbol for the mean is x

r

. It is found using the formula x

f fx =

r

/

. • The mode is the score with the highest frequency. This is the score that occurs the most oen.

• The median is the middle score when the scores are arranged in ascending order.

• The cumulave frequency(cf ) is the sum of the frequencies for all scores less than or equal to that score.

A group of people's height was measured (in cm) and the results were wrien in this table

Score ( x ) Frequency ( f ) Cumulave

frequency (cf ) fx 110 3 3 3#110 = 330 112 5 3+5=8 5#112 = 560 113 10 8+10=18 10#113 = 1130 115 9 18+9=27 9#115 = 1035 116 8 27+8=35 8#116 = 928 35 f =

/

_fx =3983 f

/

means 'sum of frequencies'

f# x cm 113.8 x f fx 35 3983 = = =

r

/

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Interpreting Data

Basics

d

Frequency Histograms and Polygons

A histogram is a column graph based on data. A polygon is made up of straight lines joining the centres of these columns.

Find the mean (to1 decimal place). Answer the quesons about this diagram

Frequency Histogram and Polygon

F r e q u e n c y ( f ) Score ( x) 10 9 8 7 6 5 4 3 2 1 0 2 3 4 5 6 7 8 9 10 Centres of columns are joined Polygon Histogram

Leave half a column on either side

a _{What is the frequency of the score} _6?

9 (from the histogram)

b _{What is the cumulave frequency of scoring a}_6?

of frequency of frequency of frequency of

2 6 9

17

cf 6 = 2+ 4+ 6

= + +

=

c _{How many scores are there?}

f 2 6 9 7 8 3 35 = + + + + + =

/

d.p. 6.7 (1 ) x f fx 35 2 2 6 4 9 6 7 7 8 9 3 10# # # # # # = = + + + + + =

r

/

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Interpreting Data

Basics

Cumulative Frequency Histograms and Polygons

A cumulave frequency histogram can also be joined. The polygon joins the right corners of the histogram. A cumulave frequency polygon is also called an "ogive".

Answer these quesons about the diagram below

a

b

c

d

What is the cumulave frequency of the score6?

What is the frequency of scoring a 6?

How many scores are there?

Find the median.

Since there were 10 scores, the median is the average of the two middle scores (in posion 5 and6) Take the cumulave frequency of the last score. This means that there are 10 scores in total.

5 (from the histogram)

frequency of 6 cumulative frequency of 6 cumulative frequency of 5

5 3

2

=

-=

median score in 5 position score in 6 position

. 2 2 6 7 6 5 th th ` = + = + =

Cumulave Frequency Histogram and Polygon

C u m u l a t v e f r e q u e n c y ( f c ) Score ( x) 10 9 8 7 6 5 4 3 2 1 0 2 3 4 5 6 7 8 9 10 Right corners of columns are joined

Cumulave frequency polygon

Cumulave

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Interpreting Data

Questions

Basics

1. A group of people were asked how many languages they speak and this table was partly completed.

a

b

c

d

e

Complete the table.

Show that the mean is_x

_r

=2 25. .

What is the median?

What is the mode?

Are

/

f and the cumulave frequency of x= 5 equal? Why?

Number of languages( x ) Frequency( f ) fx Cumulave frequency(cf )

1 20 20#1 = 20 2 38 3 12#₃₌₃₆ ₅₀ 4 7 57 5 3 60 f =

/

fx=

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Interpreting Data

Questions

Basics

2. A group of people were asked how many movies they had seen in the last year. The diagram below shows the frequency polygon for the results.

What is the mean (to 2 decimal places if necessary)?

What is the median?

What is the mode?

Number of movies seen

F r e q u e n c y ( f ) Movies ( x) 9 8 7 6 5 4 3 2 1 0 20 21 22 23 24 25

Movies( x ) Frequency( f ) fx Cumulave frequency(cf )

f =

/

fx= a b c d

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Interpreting Data

Questions

Basics

3. A group of people were asked their age and this frequency histogram was produced.

Complete the table below.

Complete the polygon on the diagram. Find the mean age (to 2 d.p. if necessary).

What is the mode?

What is the median?

Ages( x ) Frequency( f ) fx Cumulave frequency(cf )

f =

/

fx= a b c d e Ages ( x) Dierent Ages F r e q u e n c y ( f ) 6 5 4 3 2 1 0 30 31 32 33 34 35

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Interpreting Data

Knowing More

The median is the middle score of the data (or the average of the two middle scores). This means that 50% of the scores are less than or equal to the median. Quarles work the same way.

There are 3 quarles.

Quartiles

• The rst quarle – wrien as Q1 – is the score that25% of the scores are less than or equal to. Q1is

the median of the lower half of the scores.

• The second quarle – wrien as Q2 – is the median.

• The third quarle – wrien as Q3 – is the score that75% of the scores are less than or equal to. Q3 is

the median of the upper half of the scores.

Find the lower quarle Q1

Find the upper quarle Q3?

Since there are20 scores, the Q1 will be the average of the scores in the 5th and 6th posions. From the

table, the score in 5th posion is 2, and the score in the 6th posion is2.

c d Q 2 3 3 3 2 ` = + = 4 5 _. Q 2 4 5 3 ` = + =

Since there are20 scores, the Q3 will be the average of the scores in the 15th and16th posions. From the

table, the score in the15th_{posion is} _{4, and the score in the}16th_{posion is}_5.

How many scores are there in total?

What is the median(Q2)?

There are20 scores since f _/ = 20

The median will be the average of the two middle scores. That is the average of the scores in 10th_and₁₁th

posion. Since the cf of x= 2 is7 and the cf of x = 3 is13, so the scores in10th_and₁₁th_{posion are both} _3.

a b 2 2 Q 2 2 1 ` ₌ + ₌

In other words the quarles divide the data into quarters.

Range of scores Lowest value

Lower quarle Upper quarle

Median Highest value

Answer the following quesons about this table of data

Score ( x ) Frequency ( f ) Cumulave frequency (cf )

1 3 3 2 4 7 3 6 13 4 2 15 5 1 16 6 4 20 f =

/

20 Q1 Q3

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Interpreting Data

Knowing More

A5-point summary of a data set is a list of : The lowest value; Q1; Q2; Q3 and the highest value. Here is an example:

• The range of a data set is the dierence between the highest score and the lowest score.

• The interquarle range is the dierence between the upper and lower quarles. It is wrien as IQR.

Range and Interquartile Range

Answer the quesons about the table below

Find Q1

Find Q2

Find Q3

Find the range and interquarle range?

Write down a5-point summary for this set of data

• Lowest score= 10 • Q1= 12 (lower quarle)

• Q2= 16 (the median)

• Q3= 17 (upper quarle)

• Highest score= 20

There are 36 scores in total, so Q1 is the average of the 9th and 10thscores (median of the lower half). The

cf of x= 10 is5 and the cf of x= 12 is13.

There are 36 scores in total, so the median is the average of the scores in18th_and₁₉th_posion:

There are 36 scores in total, so Q3 is the average of the in 27th and the28th scores. The cf of x= 16 is 27

and cf of x= 18 is 33:

a

b

c

d

e

Score ( x ) Frequency ( f ) Cumulave frequency (cf )

10 5 5 12 8 13 14 4 17 16 10 27 18 6 33 20 3 36 36 f =

/

IQR Q 3 Q1 ` = -12 -12 Q 2 12 1 ` ₌ + ₌ Q 2 16 16 ₁₆ 2 ` = + = Q3 16 ₂18 17 ` = + =

Range Highest Score Lowest Score

20 10 10 = -= - = IQR Q Q 17 12 5 3 1 = -= - =

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Interpreting Data

Questions

Knowing More

1. Here is a list of data:5 , 7 , 2 , 3 , 8 , 0 , 1 , 2 , 4 , 8 , 4 , 2 , 7 , 8 , 0 , 1.

Arrange this data into ascending order.

Find the median of this data set.

Find Q1.

Find the upper quarle.

Write down a5-point summary of this data set.

What is the range of this data set?

What is in the interquarle range of this data set?

a b c d e f g

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Interpreting Data

Questions

Knowing More

2. A group of students' exam results are displayed in the table below.

Complete the cumulave frequency column.

How many students were used to create the table?

In which posion is the median? What is the median?

Find Q1 and Q3.

Find the interquarle range.

Write a5-point summary on this data.

a b c d e f

Results in% ( x ) Number of students( f ) Cumulave frequency(cf )

10 5 20 3 30 1 40 8 50 3 60 9 70 6 80 3 90 8 100 6 52 f =

/

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Interpreting Data

Using Our Knowledge

A5-point summary is used to plot a "Box-and-Whisker" plot for a set of Data. These are used to compare dierent data sets. They are drawn like this:

They are always drawn against a number line.

29 , 26 , 30 , 22, 30 , 21, 22 , 22 , 25 , 24 , 21 , 26

Box-and-Whisker Plots

Answer these quesons about this set of data

21 , 21, 22, 22, 22 , 24 , 25 , 26 , 26 , 29 , 30 , 30

Arrange this data into ascending order:

a

Median/Q2

20 21 22 23 24 25 26 27 28 29 30

Find a5-point summary.

Draw a box-and-whisker plot for this set of data.

b

c

There are 12 numbers in the data set. • The lowest number is21.

• Q1 will be the average of the 3rd and the4th posion so:

• The median is the average of the numbers in6th_and₇th_posion.

• Q3 will be the average of the 9rd and the10th posion so:

• The greatest number in the set is30. Q 2 22 22 ₂₂ 1 ` ₌ + ₌ . Q 2 24 25 24 5 2 ` = + = 26 29 _. Q 2 27 5 3 ` = + = Q1 Lowest value

Whisker Box Whisker

Median

Highest value

Q1 Q3

Q3

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Interpreting Data

Using Our Knowledge

In any Box-and-Whisker plot the diagram represents:

Box-and-Whisker plots are used to compare dierent sets of data.

Find a5-point summary for the temperatures in 2009 and 2010.

25% of data

25% of data 25% of data

Middle50% of data (Interquarle range)

25% of data

The table below shows average temperatures for a city over two years

a

Which year had the greater interquarle range?

It can be seen that the box-and-whisker plot for2009 has a longer interquarle range (the 'box' part) than2010. This means2009 had a greater interquarle range in temperature.

Draw box and whisker plots for the average temperatures in 2009 and2010.

Which year had the greater range of temperature?

It can be seen that the box-and-whisker plot for2010 is longer than that of 2009. This means that2010 had the greater range in temperature.

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 2009 2010 b c d

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

2009 28 22 24 22 16 26 20 25 29 20 23 24 2010 21 24 26 25 24 25 24 25 30 27 28 36 2009 2010 Ascending Order 16,20, 20, 22, 22, 23, 24, 24, 25, 26, 28, 29 21, 24, 24, 24, 25, 25, 25, 26, 27, 28, 30, 36 Lowest Temperature 16 21 Q1 21 24 Q2 23.5 25 Q3 25.5 27.5 Highest Temperature 29 36

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Interpreting Data

Using Our Knowledge

Standard Deviation

Standard deviaon measures the average distance each score is away from the mean. It has this symbol using lower case sigma v n, pronounced 'sigma-n'. This is the formula forv n:

Where x

r

is the mean and

/

sll means 'sum of'. Here is an example: • Find the mean, x

r

:

• Draw a table with these3 columns.

• Use the formula for standard deviaon.

v n x x n 2 =

/

^

-

r

h

Find the standard deviaon (correct to 1 decimal place) of this set of data:11,8,13,3,9, 15,17,17,6, 11

sum of scores 11 x n 10 110 = = =

r

Score ( x ) x - rx _^x _-_rx _h2 11 11-11=0 0 8 8-11=-3 9 13 13-11=2 4 3 3-11=-8 64 9 9-11=-2 4 15 15-11=4 16 17 17-11=6 36 17 17-11=6 36 6 6-11=-5 25 11 11-11=0 0 0 x - x_r = ^ h

/

_^_x- _x_r_h2=194

This is called "mean dierence"

This total will ALWAYS be zero. If not, a mistake has been made.

"mean dierence" is squared

d.p. v . n x x 10 194 4 4 1 n 2 = -= =

r

^

h

/

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Interpreting Data

Questions

_{Using Our Knowledge}

1. An athlete runs the same race16 mes. This is how long it takes him (in seconds) to run each me: 14 ,12 ,18 ,14 , 16 ,18 ,19 ,14 ,16 ,17 ,15 ,13 ,20 ,16 ,14 ,19

11 12 13 14 15 16 17 18 19 20 21

Write these mes in ascending order.

Find a5-point summary of this data.

Draw a box-and-whisker plot for this data.

What is the range of the data?

What is the interquarle range of the data?

a

b

c

d

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Interpreting Data

Questions

Using Our Knowledge

2. During8 days, a cat and a dog eat an amount of food (in grams) according to the table below.

Mon Tues Wed Thur Fri Sat Sun Mon

Cat 70 100 40 90 50 70 55 100

Dog 65 100 90 80 75 85 50 85

Arrange each set of data into ascending order

Find a5-point summary for each data set.

Draw box-and-whisker plots for the dierent data sets on the number line below.

From the box-and-whisker plot, which had the greater interquarle range? Find the interquarle range.

a

b

c

d

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Interpreting Data

Questions

_{Using Our Knowledge}

3. 150 men and 150 women were in a survey and these are the resulng box-and-whisker plots from their ages:

How old was the youngest woman in the survey?

How old was the oldest man in the survey?

What is the median age for the women?

Find Q1 for the men.

Find Q3 for the women.

What is the age that half the men were older than?

a b c d e f Women Men 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58

(21)

Interpreting Data

Questions

Using Our Knowledge

4. Ava counted the number of books she read each month for a year. She wrote them in the table below:

Jan Feb _Mar _Apr _May Jun _Jul _Aug _Sep _Oct _Nov _Dec

5 6 4 1 6 1 9 9 10 10 5 6

Find x

r

, the mean.

Complete the table below for the above set of data.

What is the formula to calculate v n?

Show that the standard deviaon of the above set of data to 2 decimal places is2.97.

a b c d x x -_rx ^ x - rx h2 5 -1 6 0 4 1 -5 6 1 -5 9 9 3 9 10 4 10 5 6 0 0 x - xr = ^ h

/

^ x - xrh2=

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Interpreting Data

Thinking More

A set of data can be one of three things: a normal distribuon, skewed to the right or skewed to the le.

This is only a general rule of thumb which holds most of the me and excepons to this rule_{do occur. This can be}

used to compare dierent data sets.

normal distribuon skewed to the right skewed to the le

Find the median of both data sets

Find the mean of both data sets and comment on the skewness of each data set.

This is a normal distribuon since mean= median This is skewed to the right since median1 mean Data Set1: Data set1: Mean= 27.5 _Mean= 23 3 1 Data Set2: Data set2: b c

Skewness of Data

• le side = right side

• median = mean

• bell shaped

• right side is longer

• median1 mean

• not bell shaped

• le side is longer

• median2 mean

• not bell shaped

median . 2 25 30 27 5 = + = median . 2 15 20 17 5 = + =

Write both data sets in ascending order

Data Set1 (white):5 , 15 , 15, 15 , 20, 25, 30 , 35 , 35 , 40 , 45, 50

Data Set2 (black): 5 , 10 , 10 , 15, 15 , 15, 20 , 25 , 30 , 40, 45 , 50 a

Two data sets are shown on the column graph below, data set1 (white) and data set2 (black).

60 50 40 30 20 10 0

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Interpreting Data

Thinking More

Spread of Data

Thespread of a data set measures how consistent (close to the mean) a data set is. This depends on:

Strategy1: 34,35,28,28,30,31,32,27 Strategy2: 1,13,5,10,16,14,1 ,5

• Range – the wider the range of the data set the less likely scores will be close to the mean

• Interquarle range – the wider the range of the data set the less likely scores will be close to the mean • Standard deviaon – v n measures how far the scores are from the mean.

A gameplayer tries two strategies for playing a game. He tries each strategy eight mes and these are the points received

Find a5-point summary for each strategy.

a Strategy 1 Strategy 2 Lowest 27 1 Q1 28 ₂ 3 1 +5 = Q2 30.5 7.5 Q3 33 2 13.5 13 +14 = Highest 35 16

As expected, the standard deviaon for Strategy2 is greater.

Which strategy do you expect to have a larger standard deviaon?

Find the standard deviaon of both strategies to2 decimal places.

Strategy2 has a larger range and interquarle range, so it is expected to have a larger standard deviaon.

d

e

Find the range and interquarle range for each strategy.

b

Strategy 1 Strategy 2

Range 35 - 27 = 8 16 - 1 = 15

IQR 33 - 28 = 5 13.5 - 3 = 10.5

Draw a box-and-whisker plot for each strategy.

Strategy 1 Strategy 2 c 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 v n for Strategy1:2.74 (2 d.p.) v n for Strategy2:5.53 (2 d.p.) v n x x n 2 =

/

^

-

r

h

(24)

Interpreting Data

Questions

_{Thinking More}

1. Two movies received reviews from eight crics who gave the movie a score between1 and10. Here are their results:

Use this table to nd the standard deviaon of the Movies' scores:

a Movie1 Score( x ) _x-_x

_r

^

x _-x

r

h

2 3 8 8 6 3 5 2 5 x x /

^

-

r

h

= x x 2 /

^

_-

_r

h

₌ Movie2 Score( x ) _x-_x

_r

^

x _-x

r

h

2 8 7 3 4 7 10 6 3 x x /

^

-

r

h

= x x 2 /

^

_-

_r

h

₌

Complete these tables for the movies:

How are the scores for each movie skewed?

Which movie has the more consistent scores?

b c d Movie1 Movie2 Lowest Q1 Q2 Q3 Highest vn Movie1: 3 , 8 , 8 , 6 , 3 , 5 , 2 , 5 Movie2: 8 , 7 , 3 , 4 , 7 , 10 , 6 , 3

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Interpreting Data

Questions

Thinking More

2. At the olympics, divers receive a score between1 and10 each me they dive. These are the scores aer 12 dives for the divers who came in rst and second place.

DiverA DiverB

7 , 8 , 5 , 7 , 8 , 6 , 6 , 5 , 8 , 5 , 8 , 5 7 , 6 , 4 , 5 , 6 , 5 , 10 , 9 , 8 , 6 , 9 , 9

Find a5-point summary for each diver's scores

Find the range and interquarle range of each diver's scores

Draw a box-and-whisker plot for each of the divers' scores. Are the scores skewed?

a

b

c

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Interpreting Data

Questions

_{Thinking More}

Draw up a table with the headings, score( x ), x -x

_r

and x

^

-x

r

h

2 and use it to nd v n for both divers.

If the winner is based on the total scores, which diver won?

Which diver had more consistent scores?

d

e

(27)

Interpreting Data

Questions

Thinking More

3. Answer these quesons about skewness of data:

If the median is less than the mean which way would the data be skewed (according to the rule of thumb)?

When is data skewed to the le (according to the rule of thumb)?

Sketch a box-and-whisker plot represenng data that is skewed to the le.

Standard deviaon is a measure of how far each score is from the mean. What does this mean?

How is the standard deviaon aected by the consistency of the scores?

Does data with a higher or lower standard deviaon have more consistency?

a b c d e f

(28)

Interpreting Data

Answers

Basics:

1. 2. 3. a b c d

e _Yes,

/

_f_{and the cumulave frequency of}

x = 5 are equal as x = 5 is the highest score in the data. This means all the scores are less than or equal to5.

N u m b e r o f l a n g u a g e s ( x ) F r e q u e n c y ( f ) f x C u m u l a  v e f r e q u e n c y ( c f ) 1 2 0 2 0 # 1 = 2 0 0 + 2 0 = 2 0 2 3 8 - 2 0 = 1 8 1 8 # 2 = 3 6 2 0 + 1 8 = 3 8 3 5 0 - 3 8 = 1 2 1 2 # 3 = 3 6 3 8 + 1 2 = 5 0 4 7 4 # 7 = 2 8 5 0 + 7 = 5 7 5 3 5 # 3 = 1 5 5 7 + 3 = 6 0 6 0 f =

/

2 0 3 6 3 6 2 8 1 5 1 3 5 f x = + + + + =

/

median₌ ₂ 2.25 x f fx 60 135 = = =

r

/

mode= 1 M o v i e s ( x ) F r e q u e n c y ( f ) f x C u m u l a  v e f r e q u e n c y ( c f ) 2 0 8 2 0 × 8 = 1 6 0 8 2 1 5 2 1 × 5 = 1 0 5 8 + 5 = 1 3 2 2 7 2 2 × 7 = 1 5 4 1 3 + 7 = 2 0 2 3 3 2 3 × 3 = 6 9 2 0 + 3 = 2 3 2 4 2 2 4 × 2 = 4 8 2 3 + 2 = 2 5 2 5 4 2 5 × 4 = 1 0 0 2 5 + 4 = 2 9 f =

/

2 9 f x =

/

6 3 6 a b c d ( d.p.) 21.93 2 x

_r

= median ₌ ₂₂ mode= 20 g e s x ) F r e q u e n c y ( f ) f x C u m u l a  v e f r e q u e n c y ( c f ) 3 0 4 3 0 × 4 = 1 2 0 4 3 1 3 3 1 × 3 = 9 3 4 + 3 = 7 3 2 3 3 2 × 3 = 9 6 7 + 3 = 1 0 3 3 3 3 3 × 3 = 9 9 1 0 + 3 = 1 3 3 4 2 3 4 × 2 = 6 8 1 3 + 2 = 1 5 3 5 5 3 5 × 5 = 1 7 5 1 5 + 5 = 2 0 f =

/

2 0 f x =

/

6 5 1 a

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Interpreting Data

Answers

Basics:

Knowing More:

Using Our Knowledge:

3. 1. 1. 2. c mode= 35 d e b 3 . x

_r

= 2 55 median= 3 .2 5 a b 0 , 0 , 1 , 1 , 2 , 2 , 2 , 3 , 4 , 4 , 5 , 7 , 7 , 8 , 8 , 8 median= 3.5 c _Q1 = 1.5 • Lowest score = 0 • Q1 = 1.5 (lower quarle) • Q2 = 3.5 (the median) • Q3 = 7 (upper quarle) • Highest score= 8 d e Q3 = 7 f Range =8 g

The total number of students used is 52

a b Results in% ( x ) Number of students₍_f₎ Cumulave frequency(cf ) 10 5 5 20 3 5 + 3 = 8 30 1 8 + 1 = 9 40 8 9 + 8 = 17 50 3 17 + 3 = 20 60 9 20 + 9 = 29 70 6 29 + 6 = 35 80 3 35 + 3 = 38 90 8 38 + 8 = 46 100 6 46 + 6 = 52 52 f =

/

. IQR= 5 5 c median= 60 Q1 = 40 Q3 = 90 d e f _• _{Lowest score}= 10 • Q1 = 40 (lower quarle) • Q2 = 60 (the median) • Q3 = 90 (upper quarle) • Highest score= 100 IQR = 50 12 , 13 , 14 , 14 , 14 , 14 , 15 , 16 , 16 , 16 , 17 , 18 , 18 , 19 , 19 , 20 • Lowest score= 12 a b • Q1 = 14 (lower quarle) • Q2 = 2 16 +16 ₌ ₁₆ (the median) • Q3 = 18 (upper quarle) • Highest score= 20 Ages( x) Dierent Ages F r e q u e n c y ( f ) 6 5 4 3 2 1 0 30 31 32 33 34 35

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Interpreting Data

Answers

Using Our Knowledge:

1. 2. 2. 3. c d e Range =8 IQR= 4 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 Cat 40 50 55 70 70 90 100 100 Dog 50 65 75 80 85 85 90 100 a • Lowest score= 40 • Q1 = 52.5 (lower quarle) • Q2= 70 (the median) • Q3 = 95 (upper quarle) • Highest score= 100 • Lowest score= 50 • Q1 = 70 (lower quarle) • Q2= 82.5 (the median) • Q3 = 87.5 (upper quarle) • Highest score= 100 b _Cat: Dog: a b c d e f

From the box-and-whisker plot the youngest woman was24 years old. The oldest man was 56 years old

The median age for the women was 38 years old

Q1 for the men was39 years old

Q3 for the women was46 years old

The age that half the men were older than is the median. The median is 46 years old.

c 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0 8 5 9 0 9 5 1 0 0 D o g C a t d IQR (dog) = 17.5 IQR (cat) = 42.5

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Interpreting Data

Answers

a b c d x x -_rx ^ x -x rh2 5 -1 _{( )}₁2 ₁ = 6 0 0 4 4 - 6 = -2 4 1 -5 25 6 6 - 6 = 0 0 1 -5 25 9 9 - 6 = 3 9 9 3 9 10 4 16 10 4 16 5 -1 1 6 0 0 x _-x

_r

₌0

^

h

/

_^

x - x

_r

_h

2 = 106

Using Our Knowledge:

Thinking More:

4. 1. 1. 6 x

_r

= / v ( ) n x x n 2 =

r

/ v ( ) 2.97 n x x 12 106 n 2 =

r

= = a a Movie1 Score ( x ) x -x

r

^

x -x

r

h

2 3 3 - 5 = -2 4 8 8 - 5 = 3 9 8 8 - 5 = 3 9 6 6- 5 = 1 1 3 3 - 5 = -2 4 5 5 - 5 = 0 0 2 2 - 5 = -3 9 5 5 - 5 = 0 0 0 x - x

_r

=

^

h

/

^

_x- _x

_r

h

2= 36 Movie2 Score ( x ) x -x

r

^

x -x

r

h

2 8 8- 6 = 2 4 7 7- 6 = 1 1 3 3 - 6 = -3 9 4 4 - 6 = -2 4 7 7- 6 = 1 1 10 10 - 6 = 4 16 6 6- 6 = 0 0 3 3 - 6 = -3 9 0 x - x

_r

=

^

h

/

^

x - x

r

h

2 = 44 Movie1: v = 2.12 Movie1: v_n = 2.35 For Movie2:

the mean= 6 and the median= 6.5, median2 mean

` _{the scores are skewed to the le (ie} there are more scores that are less than the median than there are scores greater than the median)

b c Movie1 Movie2 Lowest 2 3 Q1 3 3.5 Q2 5 6.5 Q3 7 7.5 Highest 8 10 vn 2.12 2.35

the mean= the median= 5 ` _{the data is not skewed and the} scores are distributed normally For Movie1:

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Interpreting Data

Answers

Thinking More:

1. 2.

2.

Movie 1 has the more consistent scores than Movie 2. This is because the

standard deviaon of the scores for Movie 1 is less than for Movie 2.

d Movie 1 Movie 2 Range 8 - 2 = 6 10 - 3 = 7 Interquarle range 7 - 3 = 4 7.5 - 3.5 = 4 vn 2.12 2.35

The scores for Diver A are not skewed. The scores for Diver B are skewed to the rig ht (the box-and-whisker plot is longer on the right hand side of the median)

a b Diver A Diver B Lowest 5 4 Q1 5 5.5 Q2 6.5 6.5 Q3 8 9 Highest 8 10 Diver A Diver B Range 8 - 5 = 3 10 - 4 = 6 Interquartile range 8 - 5 = 3 9 - 5.5 = 3.5 d Diver A x x x

r

( x x

r

)2 5 -1.5 2.25 5 -1.5 2.25 5 -1.5 2.25 5 -1.5 2.25 6 -0.5 0.25 6 -0.5 0.25 7 0.5 0.25 7 0.5 0.25 8 1.5 2.25 8 1.5 2.25 8 1.5 2.25 8 1.5 2.25 0 x - x

_r

=

^

h

/

( x - x

r

) 2= 19 Diver A: v_n= 1.126 c D i v e r A D i v e r B 3 4 5 6 7 8 9 1 0

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Interpreting Data

Answers

Diver A had the more consistent scores. This is shown by the lower standard deviaon of Diver A’s scores.

e f Driver B: v_n = 1.87 2. 3. 3. d Diver B x x x

r

( x x

r

)2 4 -3 9 5 -2 4 5 -2 4 6 -1 1 6 -1 1 6 -1 1 7 0 0 8 1 1 9 2 4 9 2 4 9 2 4 10 3 9 ( x - x

r

) = 0

/

( x - x

_r

) 2= 42 If the winner was based on total score, Diver B won.

When median2 mean

The data is skewed to the right.

The standard deviaon measures how spread out the scores are. The larger a standard deviaon is, the further the scores are from the mean.

The more consistent the scores, the lower the standard deviaon. The less consistent the scores, the higher the standard deviaon.

Data with a lower standard deviaon has more consistency. a b c d e f L e  s i d e i s l o n g e r

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