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# Lesson 7 measures of dispersion part 2

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### Lesson 7 measures of dispersion part 2

1. 1. Introduction to Statistics for Built Environment Course Code: AED 1222 Compiled by DEPARTMENT OF ARCHITECTURE AND ENVIRONMENTAL DESIGN (AED) CENTRE FOR FOUNDATION STUDIES (CFS) INTERNATIONAL ISLAMIC UNIVERSITY MALAYSIA
2. 2. Lecture 9 Measures of variability/dispersion Part II Today’s Lecture:  The average absolute deviation  The Standard deviation
3. 3. 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.
4. 4. 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.
5. 5. 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.
6. 6. 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.
7. 7. 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.
8. 8. Measure of central tendency Absolute deviation Mean = 5 Median = 3 Mode = 2 The absolute deviation cont.
9. 9. 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
10. 10. 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 10 434 2654 x 434 14 = = 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
11. 11. 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) - 8455² 150(150-1) 4136725 22350 185.09 = = = s = 13.6
12. 12. 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 DISPERSION Group Mean Mode Median Min. Max. Range SD A 70 67 70 40 90 50 14 B 60 55 62 55 68 13 2