IV. Measures of Variability (ppt) PDF

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Descriptive Measures. Measures of variability: standard deviation and variance...

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MEASURES OF VARIABILITY

I

OBJECTIVES

•

To realize the importance of the measure of variability in describing the characteristics of a set of data

MEASURES OF RELATIVE POSITION

•

It is a measure whether a value is about the average, or whether its unusually high or low.

•

Quantiles are statistics that describe various subdivisions of a frequency distribution into equal proportions.

•

Three special quantiles:

1.

Quartiles

2.

Deciles

3.

Percentiles

QUARTILES -measures of location that divide the ordered data into four equal parts

For UNGROUPED DATA

For GROUPED DATA

where LB = lower limit boundary of the quartile class

If the resulting positioning point is not an integer, use linear interpolation.

𝑛 = number of observations/total frequency k = quartile position 𝑐𝑓 = cumulative frequency before the quartile class f= frequency of the quartile class cw= class width

INTERQUARTILE RANGE

The interquartile range (IQR) is a measure of where the bulk of the values lie.

IQR = Q3 – Q1

DECILES -measures of location that divide the distribution in ten equal parts

For UNGROUPED DATA

For GROUPED DATA

where LB = lower limit boundary of the decile class 𝑛 = number of observations/total frequency

If the resulting positioning point is not an integer, use linear interpolation.

k = decile position 𝑐𝑓 = cumulative frequency before the quartile class f= frequency of the decile class cw= class width

PERCENTILES -measures of location that divide the distribution in one hundred equal parts

For UNGROUPED DATA

For GROUPED DATA

where LB = lower limit boundary of the percentile class

If the resulting positioning point is not an integer, use linear interpolation.

𝑛 = number of observations/total frequency k = percentile position 𝑐𝑓 = cumulative frequency before the quartile class f= frequency of the percentile class cw= class width

EXAMPLE 1: Suppose the following data are the scores of 12 students from a Statistics Test. Find the   

40, 34, 45, 48, 30, 50, 29, 33, 38, 44, 42, 35

EXAMPLE 1:



 () 

EXAMPLE 1: 

 

()

() 







The first quartile is 33.25.

EXAMPLE 1: 

 

IQR = Q3 – Q1

()

() 







IQR = Q3 – Q1 = 44.75 – 33.25 = 11.5

EXAMPLE 1:



 () 

EXAMPLE 1: 



()

() 



𝐷 = 5𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 + 𝑑𝑒𝑐𝑖𝑚𝑎𝑙 𝑣𝑎𝑙𝑢𝑒(6𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 − 5𝑡ℎ 𝑣𝑎𝑙𝑢𝑒) 



The fourth decile is 35.6.

EXAMPLE 1:



 () 

EXAMPLE 1: 29, 30, 33, 34, 35, 38, 40, 42, 44, 45, 48, 50

𝑃 : 

() 

() 

𝑃 = 7𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 + 𝑑𝑒𝑐𝑖𝑚𝑎𝑙 𝑣𝑎𝑙𝑢𝑒(8𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 − 7𝑡ℎ 𝑣𝑎𝑙𝑢𝑒) 



The 55th percentile is 40.3.

EXAMPLE 2 The data below is the age of the residents in Barangay 123. Compute 





Classes 55-59 50-54 45-49 40-44 35-39 30-34 25-29

f 55 26 37 37 48 42 27

EXAMPLE 2 Solution: Columns for lower boundary (LB) and for cumulative frequency (cf): Classes

f

55-59 50-54 45-49 40-44 35-39 30-34 25-29

55 26 37 37 48 42 27 

 𝑓 = 𝑛 = 272 

LB 54.5 49.5 44.5 39.5 34.5 29.5 24.5

cf 272 217 191 154 117 69 27

QUARTILE CLASS 

the class containing the (  )th item

EXAMPLE 2 The class containing the

()()

Classes 55-59 50-54 45-49 40-44 35-39 30-34 25-29



= 68th item is the quartile class which is the 30-34. f 55 26 37 37 48 42 27

LB 54.5 49.5 44.5 39.5 34.5 29.5 24.5

cf 272 217 191 154 117 69 27

𝑛𝑘 68 − 27 5 4 − 𝑐𝑓 𝑐𝑤 = 𝟑𝟒. 𝟑𝟖 𝑄 = 𝐿𝐵 + = 29.5 + 𝑓 42

EXAMPLE 2 The class containing the 40-44.





Classes 55-59 50-54 45-49 40-44 35-39 30-34 25-29

 ()() 

f 55 26 37 37 48 42 27

th item is the decile class which is the LB 54.5 49.5 44.5 39.5 34.5 29.5 24.5

cf 272 217 191 154 117 69 27

𝑛𝑘 − 𝑐𝑓 𝑐𝑤 136 − 117 5 10 𝐷 = 𝐿𝐵 + = 𝟒𝟐. 𝟎𝟖 = 39.5 + 37 𝑓

EXAMPLE 2 The class containing the which is the 45-49.





Classes 55-59 50-54 45-49 40-44 35-39 30-34 25-29

𝑃

 ()() 

f 55 26 37 37 48 42 27

item is the percentile class LB 54.5 49.5 44.5 39.5 34.5 29.5 24.5

cf 272 217 191 154 117 69 27

𝑛𝑘 − 𝑐𝑓 𝑐𝑤 190.4 − 154 5 100 = 𝐿𝐵 + = 𝟒𝟗. 𝟒𝟐 = 44.5 + 37 𝑓

EXAMPLE 2 

34.48

25% of the residents of Barangay 123 are aged 34.38 and below, and 75% of the residents are older than 34.38.



42.08

50% of the residents of Barangay 123 are aged 42.08 and below, and the other 50% are older than 42.08.



49.42

70% of the residents of Barangay 123 are aged 49.42 and below, and the other 30% are older than 49.42.

Measures of variability or dispersion are measures that indicate how dispersed or scattered are the data. With measures of variability, the measures of central tendency becomes more meaningful and data are better described.

MEASURES OF VARIABILITY

RANGE - the difference between the highest and lowest values in a set of data •

For ungrouped data:

•

For grouped data:









STANDARD DEVIATION

•

a measure of how far away items in a data set are from the mean

•

the larger the standard deviation, the more variation there is in the data set

•

The smallest possible value for the standard deviation is 0 and it can never be a negative number

STANDARD DEVIATION

Formula for ungrouped data:

Sample Standard Deviation: ∑(𝑥 − 𝑥 ) 𝑛−1 Population Standard Deviation. 

𝑠=

𝜎=



 ∑  (𝑥 − 𝜇) 𝑁

where 𝑥 = data values; 𝑥 or μ = mean

𝑛 or N= number of observations

STANDARD DEVIATION

Formula for grouped data:

i. Sample Standard Deviation: ∑ 𝑓(𝑥 − 𝑥 ) 𝑛−1 i. Population Standard Deviation. 𝑠=

𝜎=





 ∑  𝑓(𝑥 − 𝜇) 𝑁

where x = data values; x or μ = mean;

𝑓 = frequency

𝑛 or N= number of observations

VARIANCE

•

It represents all data points in a set and is calculated by averaging the squared deviation of each mean.

•

Note that we cannot use variance as a measure of variability because generally (unit)2 will not make sense to interpret at the end.

•

Notation:

𝑠 for sample standard deviation

𝜎 for population standard deviation 𝑠2 is sample variance 𝜎2 is population variance

EXAMPLE 3 Eight students were asked on how much they gained weight in three months of staying at home in kilograms (kg). Below are the data. Find the standard deviation and variance. Student A B C D E F G H Total

x 5 4 4.5 4 5 6 5 5 38.5

𝑠 =

∑(𝑥 − 𝑥 ) 𝑛−1

EXAMPLE 3 Eight students were asked on how much they gained weight in three months of staying at home in kilograms (kg). Below are the data. Find the standard deviation and variance. Student A B C D E F G H

x 5 4 4.5 4 5 6 5 5

(𝑥 − 𝑥 ) 0.1875 -0.8125 -0.3125 -0.8125 0.1875 1.1875 0.1875 0.1875

(𝑥 − 𝑥 ) 0.035 0.660 0.098 0.660 0.035 1.41 0.035 0.035

EXAMPLE 3 Eight students were asked on how much they gained weight in three months of staying at home in kilograms (kg). Below are the data. Find the standard deviation and variance. Student A B C D E F G H Average

x 5 4 4.5 4 5 6 5 5 4.8125

(𝑥 − 𝑥 ) 0.1875 -0.8125 -0.3125 -0.8125 0.1875 1.1875 0.1875 0.1875

(𝑥 − 𝑥 ) 0.035 0.660 0.098 0.660 0.035 1.41 0.035 0.035 Total = 2.97

EXAMPLE 3 Student A B C D E F G H Average

(𝑥 − 𝑥 ) 0.1875 -0.8125 -0.3125 -0.8125 0.1875 1.1875 0.1875 0.1875

x 5 4 4.5 4 5 6 5 5 4.8125

s

∑ (𝑥 − 𝑥 ) = 𝑛−1 =

(𝑥 − 𝑥 ) 0.035 0.660 0.098 0.660 0.035 1.41 0.035 0.035 Total = 2.97

. 

s = 0.42

𝑠 = 0.65

The data values differ from the mean (4.82 kg) on an average of about 0.65 kg.

EXAMPLE 3 Student A B C D E F G H Average

x 5 4 4.5 4 5 6 5 5 4.8125

(𝑥 − 𝑥 ) 0.1875 -0.8125 -0.3125 -0.8125 0.1875 1.1875 0.1875 0.1875

(𝑥 − 𝑥 ) 0.035 0.660 0.098 0.660 0.035 1.41 0.035 0.035 Total = 2.97

4.82 + 0.65 = 5.47 4.82 – 0.65 = 4.17 On average, the gained weight of the students in three months of staying at home fall between 4.17 to 5.47 kilograms.

EXAMPLE 4 The data below is the age of the residents in Barangay 123. Classes 55-59 50-54 45-49 40-44 35-39 30-34 25-29

f 55 26 37 37 48 42 27



 





EXAMPLE 4 The data below is the age of the residents in Barangay 123.

Classes 55-59 50-54 45-49 40-44 35-39 30-34 25-29

f 55 26 37 37 48 42 27

x 57 52 47 42 37 32 27

EXAMPLE 4 The data below is the age of the residents in Barangay 123.

Classes 55-59 50-54 45-49 40-44 35-39 30-34 25-29

f 55 26 37 37 48 42 27

x 57 52 47 42 37 32 27

fx 3135 1352 1739 1554 1776 1344 729

EXAMPLE 4 The data below is the age of the residents in Barangay 123.

Classes 55-59 50-54 45-49 40-44 35-39 30-34 25-29

f 55 26 37 37 48 42 27

Total

272

x 57 52 47 42 37 32 27

fx 3135 1196 1739 1554 1776 1344 729

𝑥 =42.75

11629

EXAMPLE 4 The data below is the age of the residents in Barangay 123.

Classes 55-59 50-54 45-49 40-44 35-39 30-34 25-29 Total

f 55 26 37 37 48 42 27

272

x 57 52 47 42 37 32 27

fx 3135 1196 1739 1554 1776 1344 729

𝑥 =42.75

11629

(𝒙𝒊 − ) 𝒙𝟐 205.91 87.41 18.92 0.42 31.93 113.43 244.94

EXAMPLE 4 The data below is the age of the residents in Barangay 123.

Classes 55-59 50-54 45-49 40-44 35-39 30-34 25-29 Total

f 55 26 37 37 48 42 27

272

x 57 52 47 42 37 32 27

fx 3135 1196 1739 1554 1776 1344 729

𝑥 =42.75

11629

(𝒙𝒊 − ) 𝒙𝟐 202.96 85.49 18.03 0.57 33.10 115.64 248.18

f(𝒙𝒊 − ) 𝒙𝟐 11162.68 2222.86 667.16 21.02 1589.03 4856.95 6700.81 27220.50

EXAMPLE 4 The data below is the age of the residents in Barangay 123. ∑ 𝑓(𝑥 − 𝑥 ) 𝑠 = 𝑛−1 27,220.5 = 272 − 1 

𝑠  = 100.44 𝑠 = 10.02

Classes 55-59 50-54 45-49 40-44 35-39 30-34 25-29 Total

f 55 26 37 37 48 42 27

272

x 57 52 47 42 37 32 27

fx 3135 1196 1739 1554 1776 1344 729

𝑥 =42.75

11629

(𝒙𝒊 − ) 𝒙𝟐 202.96 85.49 18.03 0.57 33.10 115.64 248.18

f(𝒙𝒊 − ) 𝒙𝟐 11162.68 2222.86 667.16 21.02 1589.03 4856.95 6700.81 27220.50

EXAMPLE 4 The data below is the age of the residents in Barangay 123.

Classes 55-59 50-54 45-49 40-44 35-39 30-34 25-29

f 55 26 37 37 48 42 27

𝑠 = 10.02 𝑥 =42.75

ATTENDANCE POINTS The data below is the age of the residents in Barangay 123.

Classes 55-59 50-54 45-49 40-44 35-39 30-34 25-29

f 55 26 37 37 48 42 27

𝑠 = 10.02 𝑥 =42.75 HOW TO INTERPRET s=10.02?

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