Introduction to Statistics for Built Environment

Course Code: AED 1222

Compiled byDEPARTMENT OF ARCHITECTURE AND ENVIRONMENTAL DESIGN (AED)

CENTRE FOR FOUNDATION STUDIES (CFS)INTERNATIONAL ISLAMIC UNIVERSITY MALAYSIA

Lecture 9Measures of variability/dispersion

Part II

Today’s Lecture: The average absolute deviation The Standard deviation

What is ‘deviation’?

English Dictionary: ‘Departure from a standard or norm’.

In Statistics, deviation is defined as the difference between one of a set of values and some fixed value, usually the mean of the set.

The difference between each data item and the mean of all the data items in a data set is called a deviation.

The average deviation

The average deviation is one of several measures of variability that statisticians use to characterize the dispersion among the measures in a given population/sample.

To calculate the average deviation of a set of scores it is first necessary to compute their mean, and then specify the distance between each score and that mean without regard to whether the score is above or below the mean.

The average deviation is then defined as the mean of these absolute values.

Calculating Average Deviation

1. Find the average of your measurements.

2. Find the difference between the average and each of your measurements use absolute value.

3. Find the average of these differences. This will be your deviation.

Exercise 1

The data set below indicates the age of trucks. Calculate the average (mean) absolute deviation for the data set.

18, 19, 19, 19, 19, 19, 20, 20, 45, 45, 46, 47, 48, 50.

The average absolute deviation

• The mean/average absolute deviation: is the absolute deviation from the mean.

• The average absolute deviation from the median: is the absolute deviation from the median.

• The average absolute deviation from the mode: is the absolute deviation from the mode.

Measure of central tendency

Absolute deviation

Mean = 5

Median = 3

Mode = 2

The absolute deviation cont.

The standard deviation

The standard deviation is the most important & most useful measure of spread.

Standard deviation = the positive square root of the average of the squared deviations of the individual data items about their mean.

Standard deviation = (how far away items in a data set are from their mean).

The most widely used for describing the spread of a group of scores.

Calculating standard deviation

Calculating std. deviation

Ages (x) Mean age (x) (x – x) (x – x)²

18 31 -13 169

19 31 -12 144

19 31 -12 144

19 31 -12 144

19 31 -12 144

19 31 -12 144

20 31 -11 121

20 31 -11 121

45 31 14 196

45 31 14 196

46 31 15 225

47 31 16 256

48 31 17 289

50 31 19 361

10434 2654

x 43414

= = 31Mean,

Std. deviation, s =

√√ 2654

14 - 1

Total

√ 204.15

14.29

All deviations are squared to eliminate negative values

Assuming the data is a sample

Class Boundary

(f) (m) f.m f.m²

18 – 23 2 20 40 800

23 – 28 3 25 75 1875

28 – 33 3 30 90 2700

33 – 38 5 35 175 6125

38 – 43 9 40 360 14400

43 – 48 12 45 540 24300

48 – 53 20 50 1000 50000

53 – 58 21 55 1155 63525

58 – 63 28 60 1680 100800

63 – 68 22 65 1430 92950

68 – 73 7 70 490 34300

73 – 78 9 75 675 50625

78 – 83 4 80 320 25600

83 - 88 5 85 425 36125

150Total 8455 504125

150(504125) - 8455²150(150-1)

413672522350

185.09

=

=

=

s = 13.6

Exercise

Questions:1. Which group did generally better in the exam?2. Which group had the single lowest score?3. Which group had the widest spread of scores?4. Which group had the most homogenous scores on the test?

CENTRAL TENDENCY DISPERSIONGroup Mean Mode Median Min. Max. Range SD

A 70 67 70 40 90 50 14B 60 55 62 55 68 13 2