Descriptive Statistics LECTURE N0 4.pdf
- 2. 2 MATH – 361 Introduction to Probability and Statistics Lecture No. 04 Measures of Central Tendency Reference: Ch # 1, Sec 1.2, Text Book No. of Slides: 22
- 3. 3 After completing this lecture, students will be able to Interpret measure of Central Tendency Compute different types of Mean for ungrouped data Compute different types of Mean for grouped data Desired Learning Objectives
- 4. 4 Central Tendency Definition A single value of the data which truly represents the whole data or The tendency of the observations to cluster in the central part of the data set is called Central Tendency Measure of Central Tendency A single figure that represents max data
- 5. 5 Definition Measures that describe a data set by identifying the 'central' or middlemost value by a single number usually described as ‘average’ Measure of Central Tendency A single figure that tends to lie in the middle
- 6. 6 Mathematical Average (Pythagorean Means) 1. Arithmetic Mean 2. Geometric Mean 3. Harmonic Mean Positional Measures: 4. Median 5. Mode Measure of Central Tendency
- 7. 7 Definition The sum of all the values divided by the number of values in the raw data This type of mean is called Arithmetic Mean as it involves Arithmetic Operators (+, ÷) Arithmetic Mean is denoted by Population Mean Sample Mean X Arithmetic Mean
- 10. 10 Grouped Data Find Arithmetic Mean for the given data Classes Freq (f) 65 – 84 9 85 – 104 10 105 – 124 17 125 – 144 10 145 – 164 5 165 – 184 4 185 – 204 5 Arithmetic Mean
- 12. 12 Grouped Data Classes Class Marks (x) Freq (f) fx 65 – 84 (65+84)/2 =74.5 9 9 * 74.5 = 670.5 85 – 104 94.5 10 945.0 105 – 124 114.5 17 1946.5 125 – 144 134.5 10 1345.0 145 – 164 154.5 5 772.5 165 – 184 174.5 4 698.0 185 – 204 194.5 5 972.5 Arithmetic Mean
- 13. 13 Arithmetic Mean ҧ 𝑥 = 𝑆𝑢𝑚 𝑜𝑓 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑎𝑛𝑑 𝑐𝑙𝑎𝑠𝑠 𝑚𝑎𝑟𝑘𝑠 𝑠𝑢𝑚 𝑜𝑓 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑖𝑒𝑠 = σ 𝑓𝑥 σ 𝑓 ҧ 𝑥 = σ 𝑓𝑥 σ 𝑓 = 7350 60 = 122.5 Grouped Data
- 14. 14 When to use AM The Arithmetic Mean is appropriate when all values in the data sample have the same units of measure, e.g. all numbers are heights, or dollars, or miles, etc Arithmetic Mean is a true mean if the dataset is of linear type If the dataset is of exponential type mean it involves growth rates or interests than this mean is of no use Arithmetic Mean (AM)
- 15. 15 A simple idealized example would be a dataset where each number is produced by adding 3 to the previous number: 1, 4, 7, 10, 13, 16, 19… The arithmetic mean thus gives us a perfectly reasonable middle value (1 + 4 + 7 + 10 + 13 + 16 + 19) ÷ 7 = 10 Arithmetic Mean
- 16. 16 Not all datasets are best described by this relationship. Some have multiplicative or exponential relationship, for instance if we multiplied each consecutive number by 3 rather than adding by 3 as we did above 1, 3, 9, 27, 81, 243, 729… Arithmetic Mean
- 17. 17 In this situation, the Arithmetic Mean is ill-suited to produce an “average” number to summarize this data (1 + 3 + 9 + 27 + 81 + 243 + 729) ÷ 7 AM = 156.1 Arithmetic Mean
- 18. 18 156 isn’t particularly close to most of the numbers in our dataset This skew is more apparent when the data is plotted on a flat number line Arithmetic Mean
- 19. 19 So what to do? Introducing… The Geometric Mean Since the relationship is multiplicative, to find the Geometric Mean we multiply rather than add all the numbers. Then to rescale the product back down to the range of the dataset, we have to take the root, rather than simply dividing Arithmetic Mean
- 20. 20 Practice Problem 1 Calculate Arithmetic Mean from the given grouped data Classes Frequency 2-4 3 5-7 7 8-10 9 11-13 5 14-16 4 17-19 6 Central Tendency
- 21. 21 Practice Problem 2 Calculate Arithmetic Mean for the following data 42,36,45,33,54,46,27,38,51,49,29,32 Central Tendency