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- 1. BA 301 ENGINEERING MATHEMATICS 3 TOPIC 1 : STATISTICS
- 2. Introduction Statistics is a branch of mathematics which involved the collecting , recording, arranging , studying and analyzing information consisting of numerical data. In our daily life , statistics is widely used to provide important and useful information about finance, insurance , tax, agricultural and manufacturing output, medical records, economy, import and export data, employment , education and others.
- 3. Introduction This topic explains basic terminologies of statistics. Data presentation is made in the form of graphs and frequency distribution tables. Measure of central tendency is determined using formulaic and graphical methods. This topic also discusses the calculation of mean deviation, variance and standard deviation.
- 4. Learning Outcome Know the basics of statistics Understand frequency distribution Compute measures of central tendency Understand cumulative frequency distribution Compute measures of dispersion for sample
- 5. Basics of Statistics Data Presentation Frequency Distribution Measures of Central Tendency & Dispersion
- 6. Basics of Statistics There are four components: collecting data summarizing data analyzing data presenting data
- 7. Basics of Statistics Data Collection i. ii. of information – facts or numbers Quantitative data: consist of numbers representing counts or measurements Continuous data – length, weight, mass Discrete data – Number of students Qualitative data: can be separate into different categories that are distinguished by some no numerical characteristic – genders (male/female)
- 8. Basics of Statistics Data Collection of information – facts or numbers a. Quantitative data: consist of numbers representing counts or measurements b. Qualitative data: can be separate into different categories that are distinguished by some no numerical characteristic – genders (male/female) Population Any entire collection of people, anim als, plants or things Example: Students of PTSB Sample A group of units selected from a larger group Example: Students of JKE, PTSB
- 9. Data Presentation Line Graph Bar Chart Pie Chart Histogram Ogive
- 10. Frequency Distribution The following the data represent 25 plates survey, length in mm. Arrange all data in a frequency table . 9.9 20.8 19.2 15.4 24.1 22.1 18.4 17.0 20.5 13.4 11.8 28.6 15.9 9.2 19.9 SOLUTION : i. Num of data, N = 25 ii High value = 28.6 Low value = 5.4 iii. Difference between highest value and lowest value iv. Range(Julat) = highest value – lowest value = 28.6 – 5.4 = 23.2 v. Num of class,K K = 1 + 3.3 log N = 1 + 3.3 log 25 = 5.6 ~ 6 Size of class interval , C C = Range / num of class = 23.2 / 6 = 3.866 ~ 4 15.6 12.6 16.8 12.7 19.5 8.8 23.3 5.4 14.3 7.8
- 11. FREQUENCY TABLE Class - 0.05 Class interval boundaries + 0.05 Frequency f 9.4 – 5.4 = 4 ( size of class interval ) Cumulative Frequency F 5.4 – 9.3 5.35 – 9.35 4 4 9.4 -13.3 9.35 – 13.35 4 8 13.4 – 17.3 13.35 – 17.35 7 16 17.4 – 21.3 17.35 – 21.35 6 21 21.4 – 25.3 21.35 -25.35 3 24 25.4 – 29.3 25.35 - 29.35 1 25 (9.4 – 9.3) / 2 = 0.05 25
- 12. EXAMPLE Arrange all data in a frequency table Ages of NASCAR Nextel Cup Drivers in Years (NASCAR.com) (Data is ranked--Collected Spring 2008) 21 21 21 23 23 23 24 25 25 26 26 26 26 27 27 28 28 28 28 29 29 29 29 30 30 30 30 31 31 31 31 31 32 34 35 35 35 36 36 37 37 38 38 39 41 42 42 42 43 43 43 44 44 44 44 45 45 46 47 48 48 48 49 49 49 50 50 51 51 65 72
- 13. Measure of Central Tendency Measurement Ungroup Data Mean , x x x = x N = sum of all the data N = number of data Group Data x = fx f f = sum of the value of freq x midpoint f = sum of freq
- 14. Measurement Ungroup Data Group Data Median, m - - Lm Value located at the centre of data. - Arranged in a ascending order can find from Ogive chart - Find out the located of median at the centre of data. - N 2 F fm c Lm = lower boundary of class in which the median lies. - N = total of frequency - F = cumulative freq before the class in which the median lies. - fm = frequency of the class in which the median lies. c = class size
- 15. Measurement Ungroup Data L mo Mode, mo - Value of the highest frequency - can find using Histrogram chart Group Data Find the highest frequency / often number d1 d1 d2 c Lm = lower boundary of class in which the mode lies. d1 = difference between the frequency of mode class and class before d2 = difference between the frequency of mode class and class after c = class size
- 16. Measure of Dispersion Measurement Ungroup Data xi Mean deviation , E Group Data x x xf f N xi = data x = midpoint x = mean of data x = mean of data N = number of data f = sum of freq
- 17. Measurement Ungroup Data Variance , s2 xi x 2 Group Data 2 x x f i. f N xi = data x x = midpoint = mean of data x = mean of data N = number of f = sum of freq data OR fd 2 f ii. f fd 2 f = frequency d = x (midpoint)
- 18. Measurement Standard deviation, s Ungroup Data s2 s2 = variance Group Data s2 s2 = variance
- 19. EXAMPLE Ungrouped Data Given the data : 5, 3, 6, 2, 1, 8 , 2, 2, 3, 2, 1. Find Mean, Median, Mode, Mean Deviation, Variance and Standard Deviation.
- 20. •Mean, x = = x = x N use mean formula, x bar equal total x over by number 35 11 3.182 •Median , m arrange the data to number rise and cut left and right, you have one number, there is median answer 1 ,1, 2, 2, 2, 2, 3, 3, 5, 6, 8 m=2 • Mode, m0 Recurring / often number 1, 1 , 2 , 2 ,2 ,2,3,3, 5, 6 , 8 m0 = 2 . Mean Deviation, E = xi sign negative should be neglect x n E = (1-3.182)+(1-3.182)+(2-3.182)+(2-3.182)+ (2-3.182)+ (2-3.182)+(33.182)+ (3-3.182)+ (5-3.182)+ (6-3.182)+ (8-3.182) 11 = 1.719
- 21. . Variance, s2 = xi x 2 n = (1-3.182)2+(1-3.182)2+(2-3.182)2+ (2-3.182)2+ (2-3.182)2+ (2-3.182)2 + (3-3.182)2+ (3-3.182)2+ (5-3.182)2+ (6-3.182)2+ (8-3.182)2 11 = 49.6367/11 = 4.5124 . Standard deviation, s = s2 4.5124 = 2.12

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