4.1

Measures of Central Tendency

Measures of Variability

1

• Summarizes what is average or

typical of a distribution

• Summarizes how scores are

scattered around the center of

the distribution

4.1

2

The difference between the highest and lowest scores in a distribution

• Provides a crude measure of variation

The Range

R H L

range

highest score in a distribution

lowest score in a distribution

R

H

L

4.1

3

The difference between the score at the first quartile and the score at the third quartile

• The higher the IQR, the more spread out the data points; in contrast, the smaller the IQR, the more bunched up the data points are around the mean

• Best used with other measurements such as the median and total range to build a complete picture of a data set

The Inter-Quartile Range

3 1IQR Q Q

inter-quartile range

1 the score value at or below which 25% of the cases fall

3 the score value at or below which 75% of the cases fall

IQR

Q

Q

4

4.2

5

We need a measure of variability that takes into account every score

• Deviation: the distance of any given raw score from the mean

• Squaring deviations eliminates the minus signs

• Summing the squared deviations and dividing by N gives us the average of the squared deviations

The Variance

2

2X X

sN

2

2

variance

sum of the squared deviations from the mean

total number of scores

s

X X

N

4.2

6

With the variance, the unit of measurement is squared

• It is difficult to interpret squared units

• We can remove the squared units by taking the square root of both sides of the equation

• This will give us the standard deviation

The Standard Deviation

2

X Xs

N

4.2

7

There is an easier way to calculate the variance and standard deviation

• Using raw scores

The Raw-Score Formulas

2

2 2X

s XN

2

2X

s XN

2

2

variance

standard deviation

total number of scores

mean squared

s

s

N

X

Example 4.3

Obtaining the variance and standard deviation from a simple frequency distribution

X f fX fX2

31 1 31 961

30 1 30 900

29 1 29 841

28 0 0 0

27 2 54 1,458

26 3 78 2,028

25 1 25 625

24 1 24 576

23 2 46 1,058

22 2 44 968

21 2 42 882

20 3 60 1,200

19 4 76 1,444

18 2 36 648

575 13,589

22

22

2

22

57523

25

(23) 529

13,589529 543.56 529 14.56

25

14.56 3.82

fXX

N

X

fXs X

N

fXs X

N

4.4

9

The standard deviation converts the variance to units we can understand

But, how do we interpret this new score?

• The standard deviation represents the average variability in a distribution

– It is the average deviations from the mean

• The greater the variability, the larger the standard deviation

The Meaning of the Standard Deviation

4.5

10

Used to compare the variability for two or more characteristics that have been measured in different units

• The coefficient of variation is based on the size of the standard deviation

• Its value is independent of the unit of the measurement

The Coefficient of Variation

100s

CVX

coefficient of variation

standard deviation

mean

CV

s

X

Arithmetic Mean

• Obtained by dividing the numerical values of observations by the sum of observations.

Calculating mean from grouped frequencies

Calculate the midpoint for every group

Multiply the midpoint with that group’s frequency

Add the results together

Divide the sum by number of total cases

16820/340=49.47

Grades X f fx

0 - 10 5 3 15

10 – 20 15 12 180

20 – 30 25 35 875

30 – 40 35 45 1575

40 – 50 45 110 4950

50 – 60 55 45 2475

60 – 70 65 35 2275

70 – 80 75 30 2250

80 – 90 85 15 1275

90 - 100 95 10 960

Total 340 16820

Median

The value which divides a serie of numbers in two when this series is ordered from lowest to highest.

In ungrouped data,

If n is odd (n+1)/2th value

If n is even the mean of n/2. and (n+2). values

12,24,13,46,23,15,17

• 12,13,15,17,23,24,46 n=7

• (n+1) / 2 = 4

• Med=17

12,24,13,46,23,15

• 12,13,15,23,24,46 n=6

• n/ 2 = 3 , (n+2) /2=4

• 3. eleman=15 4.eleman=23 a.o(15+23)/2=19

• Med=19

Median in grouped data

• Cumulative frequencies are calculated to determine which group contains the median

• cf= 340

• 340/2=170.th value is the median. The group containing this value is the group of the median.

Median in grouped data

l : lower limit of the median group

F : number of values lower than l

f : frequency of the median group

i : range of the median group

Grades X f C.f

0 -10 5 3 3

10 -20 15 12 15

20 – 30 25 35 50

30 – 40 35 45 95

40 – 50 45 110 205

50 – 60 55 45 250

60 - 70 65 35 285

70 - 80 75 30 315

80 - 90 85 15 330

90 - 100 95 10 340

Mode

• To find out the mode, the following formula is used after determining the group with the highest frequency.

24

Variance

• Sum of squares of deviation from mean

• General indicator for variability

26

Students IQ

Score

Student1 127 5,08 25,80 130,92 10,91

Student2 118 -3,92 15,36

Student3 125 3,08 9,48

Student4 120 -1,92 3,68

Student5 119 -2,92 8,52

Student6 125 3,08 9,48

Student7 123 1,08 1,16

Student8 120 -1,92 3,68

Student9 128 6,08 36,96

Student10 119 -2,92 8,52

Student11 120 -1,92 3,68

Student12 120 -1,92 3,68

Student13 121 -0,92 0,84

Standart Deviation

Most prominent value representing the ‘spread’ of a data set.

Most commonly used measure of variability

• LowValues are close to the mean

• High Values are farther away from the mean

Represented with a ‘s’ in short.

S=√359.31=18.96

The Coefficient of Variation

100s

CVX