Math 103 02 Central Tendency and Spread_1

7/27/2019 Math 103 02 Central Tendency and Spread_1

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Math 103Statistics andProbability

Central Tendency and Spread

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Characteristics of Data

Center: A representative or average value thatindicates where the middle of the data set is

located

Variation: A measure of the amount that the

values vary among themselves

Distribution: The nature or shape of thedistribution of data (such as bell-shaped, uniform,

or skewed) Outliers: Sample values that lie very far away

from the vast majority of other sample values

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Measures of Center

a value at the center or middle of a data set

Notation :

ΣΣΣΣ denotes the addition of a set of values

x is the variable usually used to represent theindividual data values

n represents the number of data values in a sample

N represents the number of data values in apopulation

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Mean

Mean (Arithmetic Mean) AVERAGE

the number obtained by adding the values anddividing the total by the number of values

µ is pronounced ‘myu’ and denotes the mean of all values

in a population

is pronounced ‘x-bar’ and denotes the mean of a set

of sample values

Calculators can calculate the mean of data

x = nΣΣΣΣ x

x

N µ =

ΣΣΣΣ x



2

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Mean

6.72 3 .46 3.60 6.44 26.70

Example: Find the mean of the following weights (in kg)

of sample carry-on luggages presented at an airport

check-in counter in the last hour.

Solution :

Sum of all weights = 6.72 + 3.46 + 3.60 + 6.44 + 26.70 = 46.92

Number of weights = 5

Mean = 46.92 / 5 = 9.384 kg.

Notice the impact of the outlier 26.70 on the mean.

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often denoted by (pronounced ‘x-tilde’)

or by (pronounced ‘myu-tilde’)

is not affected by an extreme value

Median

x~

Medianthe middle value when the original data values are

arranged in order of increasing magnitude

~

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Median

6.72 3.46 3.60 6.44 26.70

3.46 3.60 6.44 6.72 26.70

(odd number of values)

exact middle

MEDIAN is 6.44

6.72 3 .46 3.60 6.44

3.46 3.60 6.44 6.72

no exact middle -- shared by two numbers3.60 + 6.44

2

(even number of values)

MEDIAN is 5.02

unsorted

sorted

unsorted

sorted

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Mode

Mode

- the score that occurs most frequently- Unimodal, Bimodal, Multimodal or No Mode

- denoted by M- the only measure of central tendency that can

be used with nominal data

a. 5 5 5 3 1 5 1 4 3 5

b. 1 2 2 2 3 4 5 6 6 6 7 9

c. 1 2 3 6 7 8 9 10

Mode is 5

Bimodal - 2 and 6

No Mode



3

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Qualitative Data

376PajeroMitsubishi

581LancerMitsubishi

1,243JeepneySarao

459CRVHonda

960CityHonda

732CivicHonda

417InnovaToyota

725AltisToyota

104PriusToyota

1,098ViosToyota

Units SoldModelMaker

Mode: Sarao JeepneyCJD

Comparison

• can be used

also for nominaldata

• hardly requiresany calculation if

data is sorted

• may not exist ormay not be

unique

• not useful for

small n

• Second most

useful

• not affected by

outliers – givestruer average

• Easy to compute

if data is sorted orn is small

• varies greatlyfrom sample to

sample

• most useful

• easiest tocompute for large n

• uses all data

• does not vary

much from sampleto sample

• distribution ofmeans is well

known

• affected byoutliers

ModeMedianMean

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Weighted Mean

x =w

ΣΣΣΣ (w • x)

ΣΣΣΣ

Each individual value x may have

a weight w associated with it.

Example: A talent show is judged 40% execution,30% difficulty, 20% originality and 10% audience impact.

If a contestant scored 8,9,6 and 7, the weighted mean is

= 3.2 + 2.7 + 1.2 + 0.7 = 7.8

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Raw Data

6074745872

5882522672

6666609278

4638506650

6264686284

5466664460

8470767266

7064524078

7642506448

6440825474

Raw Data – Test Scores in a Statistics Test



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Sorted Data

9274666050

8474666050

8472666048

8272665846

8272645844

7870645442

7870645440

7668645240

7666625238

7466625026

66

64~

7.62

=

=

=

M

µ

µ

Applying the formulas, (and using a calculator) we get …

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Measures of Spread or Variation

Range

Mean Deviation

Variance

Standard Deviation

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Range and Midrange

9274666050

8474666050

8472666048

8272665846

8272645844

7870645442

7870645440

7668645240

7666625238

7466625026

Range = Highest Value – Lowest Value

In Example:

Range = 92 – 26 = 66

(a measure of spread)

Mid-Range =(92+26)/2 = 59

(a measure of center)

Mid-Range = (Highest + Lowest) / 2

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Mean Deviation

n

x x∑ −

N

x∑ − µ

Mean Dev of a Sample Mean Dev of a Population

6.72 3 .46 3.60 6.44 26.70

Example: Weights of Carry-on Luggages

Mean = 9.384 Range = 26.70 – 3.46 = 23.24

Mean Deviation =

926.65

632.34

5

384.970.26384.944.6384.960.3384.946.3384.972.6==

−+−+−+−+−



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Variance

N

x

N

i

i∑=

−

= 1

2

2

)( µ

σ

1

)(1

2

2

−

−

=

∑=

n

x x

s

n

i

i

Population Variance Sample Variance

Computing Formula for Variance

)1(

1

2

1

2

2

−

−

=

∑ ∑= =

nn

x xn

s

n

i

n

i

ii

Using n-1 will reduce the biasUsing n will underestimate variance

2

1

2

1

2

2

N

x x N N

i

N

i

ii∑ ∑= =

−

=σ

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Variance Example

824.45446.92

712.89026.70

41.4746.44

12.9603.60

11.9723.46

45.1586.72

i x

2

i x

6.72 3 .46 3.60 6.44 26.70

Example: Weights of sample Carry-on Luggages

∑

039.96)4(5

)92.46()454.824(5 2

2=

−=s

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Standard Deviation

2σ σ =

2ss =

Population SD Sample SD

In example,

800.9039.96 ==s

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Symbols for Standard Deviation

Sample Population

σ σσ σ

σσσσ x

xσ σσ σ n

s

Sx

xσ σσ σ n-1

Textbook

Some graphicscalculators

Somenon-graphicscalculators

Textbook

Some graphicscalculators

Somenon-graphics

calculators

Excel variance Excel variancevar varp



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Comparison

• Easiest to use

and interpret

• Not useful for

large n

• Says nothing

about

distribution ofdata between

max and min

• Considers all data

relative to themean

• Simple butawkward to

compute

• most useful

• Considers all datarelative to the mean

• Computation moreinvolved

• interpretation notstraight-forward

RangeMean DeviationVariance / SD

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Example : The prelim exam grades of a Statistics class

and a Calculus class are summarized below:

Coefficient of Variation

%100×= µ

σ CV

31.50%17.856.5Calculus32.26%22.068.2Statistics

Coefficientof Variation

StandardDeviation

MeanSubject

To compare spreads of samples/populations with different

means

Therefore, the statistics grades are relatively only slightly

more variable than the calculus grades.

%100×= x

sCV

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z scores

z Score (or standard score)

- A measure of position relative to other data

- the number of standard deviations that a givenvalue x is above or below the mean

Sample

z = x - x

s

Population

z = x - µσ σσ σ

Round to 2 decimal places

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Example

A student scored 67 in a calculus test and 74 in astatistics test. If the calculus test has a mean of 53 with

SD of 8, and the statistics test has a mean of 65 with SDof 6, did the student fare better relative to his classmates

in calculus or in statistics ?

Calculus: z = (67-53)/8 = 1.75

Statistics: z = (74-65)/6 = 1.50

Conclusion: The student fared better in Calculus



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Interpreting z scores

- 3 - 2 - 1 0 1 2 3Z

UnusualValues

UnusualValues

OrdinaryValues

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Measures of Location or Position

Percentiles – 100 parts (in 1%)

Deciles – 10 parts (in 10%)

Quartiles – 4 parts (in 25%)

Fractiles or Quantiles

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Percentiles

Percentiles

P1, P

2, P

3, …, P

98, P

99

i % of the data falls below (<=) Pi

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Deciles

D1, D2, D3, D4, D5, D6, D7, D8, D9

divides ranked data into ten equal parts

10% 10% 10% 10% 10% 10% 10% 10% 10% 10%

D1 D2 D3 D4 D5 D6 D7 D8 D9

i*10% of the data falls below (<=) Di

D9 is the 90th Percentile or P90



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Quartiles

Q1, Q2, Q3

divides ranked scores into four equal parts

25% 25% 25% 25%

Q3Q2Q1(minimum) (maximum)

(median)

i*25% of the data falls below (<=) Qi

Q50 is the 50th percentile P50 or 5th decile D5

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Example

9274666050

8474666050

8472666048

8272665846

8272645844

7870645442

7870645440

7668645240

7666625238

7466625026P5 =P4 =

P94=

P99=

D4 =

D9 =Q2 =

Q3 =

40 (5/100)*50 rounds up to 3rd

39 (4/100)*50=2 :get mid 38&40

83 (get mid 47th and 48th)

92 (99/100)*50 rds up to 50th

61 (4/10)*50 :get mid 20th&21st

80(9/10)*50 :get mid 45th&46th

64 (2/4)*50 :get mid 25th&26th

72 (3/4)*50 rounds up to 38th

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Decile of score x = • 10

Quartile of score x = • 4

Quantile of a Score

Percentile of score x = • 100number of scores <= x

total number of scores

number of scores <= x


number of scores <= x


If result is not an integer, Round up to the next higher integer

Example: 32 of 50 test scores are <= 66.

32/50*100=64, 32/50*10=6.4, 32/50*4=2.56

So test score 66 is in P64, D7 and Q3

To improve estimate, include only half of other scores equal to x in the numerator.

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Interquartile and Percentile Range

Percentile Range = P90 – P10

9274666050

8474666050

8472666048

8272665846

8272645844

7870645442

78706454407668645240

7666625238

7466625026 Q3 = 72

Q1 = 52IQR = 20

Range = 92 – 26 = 66

P10 = 43P90 = 80P10 to P90 Range

= 80-43 = 37

Interquartile Range (or IQR) = Q3 – Q1



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Boxplots

Simple graph to indicate Median, IQR and Outliers

Also known as the box and whisker plot

9274666050

8474666050

8472666048

8272665846

8272645844

7870645442

7870645440

7668645240

7666625238

7466625026

26 92

Q1=52 Q2=64 Q3=72 IQR=20

Variation:Whiskers extend only to 1.5*IQR below

Q1 and 1.5*IQR above Q3. Outside data

are marked with circles to mark outliers.

Basic Boxplot:Q1 Q2 Q3

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Math 103 02 Central Tendency and Spread_1