Central Tendency & Variability as 12.4.3

8/4/2019 Central Tendency & Variability as 12.4.3

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Central TendencyCentral Tendency

& Variability& Variability

A S 12.4.3


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Did you hear about thestatistician who put her

head in the oven and herfeet in the refrigerator?

She said, "On average, Ifeel just fine."


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Which average?Which average?

All three averages are useful for summarizing e.g. thedistribution of household incomes.

In 1998, the income common to thegreatest

number of households (mode) was R25000.

Half the households (median) earnedlessthan R38 885.

The mean income was R50 600.Reporting only one measure of centraltendency might be misleading and

perhaps reflect a bias.


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The table shows the heights of 50randomly chosen Grade 12 school

girls.

height (cm)midpoint (x) frequency(f) (f)(x)

150-


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height (cm) midpoint frequency(f)

150-


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It is a single summary figure that

describes the spread of data within adistribution. Range difference between thesmallest

and largest observations. Percentiles where p% of the valuesfalls

below a certain value. Interquartile Range (IQR) -Range ofthe

middle half ofmedian

scores. -

Measures of VariabilityMeasures of Variability


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The results of a survey of thetravelling

time (in minutes) of 200 workersare as

follows.Time(min)

Freq(f)

Cumfreq

0


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0 10 20 30 40 50 60

Time (min)

200

180

160

140120

100

80

60

40

20

0

Cu

m

freq

Use the graphto estimate:

the median Q2= 27

the interquartile rang

Q3=37; Q1= 16

IQR = 37 - 16=21

the no. of workers

with

no. = 200 192

= 8 workers

(100)

Q2 Q3Q1

(150)

(50)


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Unathi sells the following

number ofcomputers in 12 months:

Five-numberFive-number

summarysummaryThe summary consists of the lowest datavalue, first quartile (Q1), median (Q2), third

quartile (Q3) and highest data value.

34, 47, 1, 15, 57, 24, 20, 11, 19, 50, 28, 37

Arrange the data in ascending order.

1, 11, 15, 19, 20, 24, 28, 34, 37, 47, 50, 57


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1, 11, 15, 19, 20, 24, 28, 34, 37, 47, 50, 5

median

Median = Q2 =(24+28) /2 = 26

Q1 Q3

Q1 =(15+19) /2 = 17 Q3 =

(37+47) /2 = 42

5-number summary = 1; 17; 26; 42; 57

Give a five-number summary of the sales.

Minimum = 1 Maximum = 57


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Barry also sells computers during a 12month period.

Below is a 5-number summary for each person.

Unathi Barry

min 1 6

Q1

17 15

Q2 26 32

Q3 42 46

max 57 62

Which personwould you mostlikely want toappoint for yourcompany?

Barrys highest and lowest sales arehigher than Unathis corresponding sales,and Barrys median sales figure is also

higher than Unathis.

Barry

Explain.


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PercentilesPercentiles

A percentile is a score below which acertain percentage of values fall. Thereare 100 percentiles in a sample.

Oscars height is at the 90th

percentile andhis weight is at the 60th percentile for hisage.Describe Oscar's physical build in general

terms.

e.g. If your test score is in the 95thpercentile, it means that if 1000students took the test, at least 950students did worse than you and at

most 49 students did better than you.He is taller than 90% of thepeople but only weighs morethan 60% of the people -

possibly tall and thin.

It does not mean thatyou received 95% forthe test.


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242, 228, 217, 209, 253, 239, 266, 242,251, 240, 223, 219, 246, 260, 258, 225,234, 230, 249, 245, 254, 243, 235, 231,257.

In 2004 the snow depth at Tiffendell wasmeasured (in mm) for 25 days and

recorded.

Complete the frequency table,

cumulative frequency table and

Depth(mm)

200-210

210-220

220-230

230-240

240-250

250-260

260-270

freq

cum freq

cum %

1 2 53 7 5 2

1 3 6 1811 25234 2412 4

472

92 100


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Plot the graph of snow depthagainst the cumulative frequencyand cumulative % using two

different vertical axes.

200 210 220 230 240 250 260 270

25

20

15

10

5

0

100

80

60

40

20

0

Depth (mm)

Cum

freq

Cum

%

percentiles

Estimatethe 80th

percentile.252mm


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200 210 220 230 240 250 260 270

25

20

15

10

5

0

100

80

60

40

20

0

Depth (mm)

Cum

fre

q

Cum

%

For howmanydays wasthe depthat least250mm?

25-18= 7 days

A year later (2005)the depth is shown bythe broken line.Explain which year

had possibly better

2004 - depth greateron more days. E.g.44% below 240mmcompared to 80%

below 240mm in


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Standard deviation is useful when

comparing the spread of two or more datasets that have approximately the samemean.This technique is best used withsymmetric distributions with no outliers.

The smaller the standard deviation thenarrower the spread of measurementsaround the mean, as it has possibly fewhigh or low values.

Standard DeviationStandard Deviation

e.g. If the mean of a data set is 5 and theS.D. is 2, then on average the data lies

between 3 and 7.


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Super Crisps come in 25g bags. Thereare

two machines (A & B) producing thechips.A quality control engineer weighs a

sample

of 10 bags from each machine.

Calculate the mean of eachmachine.

A 25,3 25,6 24,8 25,7 25,5 25 24,9 25,7 25,5 25,6

B 25,3 25,3 25,4 24,9 25,3 25,3 25,4 25,4 25,4 25,3

For A: mean =

253,6

/10= 25,36g

For B: mean = 253 /10

= 25,3g


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Super Crisps will be taken to court if it is

found their bags are less than 25g.Which machine gives the best chance ofavoiding this fate?

Below is the variance and standarddeviation for each machine.

Explain.

Mean Variance S.D.

A 25,36g 1,044 1,02

B 25,3g 0,2 0,45

Machine B

A: Mass of chips are on average from24,34g to 26,38g. B: Mass of chips arefrom 24,85g to 25,75g. B has a narrower

spread than A. Its smallest value is very

Central Tendency & Variability as 12.4.3

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