Measure of central tendency (2)
- 1. MEASURES OF CENTRAL TENDENCY
- 2. Introduction One of the most important objectives of statistical analysis is to get one single value that describes the characteristic of the entire mass of data Such a value is called the central value or an average or the expected value of the variable The word average is commonly used in day to day conversation Average is defined as attempt to find a single figure to describe whole of figures
- 3. Objectives of averaging To get single value that describes the characteristic of the entire group Measures of central value, by condensing the mass of data in one single, enable us to get a bird `s eye view of the entire data
- 4. MEASURES OF CENTRAL TENDENCY Measures of central tendency are measures of the location of the middle or the center of a distribution. There are a number of measures of central tendency and these include; mean, the median, the mode
- 5. Measures of Central tendency A good Measure of Central tendency should have the following characteristics It should be easy to calculate and understand It should be unique and exist at all times It should consider all observations It should not be affected by extreme values It should be suitable for further mathematical manipulation
- 6. Mean This is the summation of all observations divided by the number of observations in the sample
- 7. Mean For Ungrouped data It is given by X = Σ Xi N =X1 +X2 +….+Xn N
- 8. Mean For Grouped data The mean for grouped data is given by X = Σ fXi Σf
- 9. Mean advantages/disadvantages Advantages It summarizes the entire distribution It could be processed further into the standard distribution It is unbiased/meaning it always gives us the population mean μ
- 10. Mean Disadvantages It may be some distance from the majority of observations Can be misleading It is approximated for grouped data Sometimes the figure obtained is not anywhere in the distribution. Can give fractional values even for ungrouped data
- 11. Properties of the Mean 1.The product of the mean and number of the values on which the mean is based is equal to the sum of all given value e.g. if the variables are 3,5,7,9 and we substitute these by the mean, the total is 24
- 12. Properties of the Mean 2.The algebras sum of the deviations of the values from the arithmetic mean is equal to zero i.e. Σ (x- x) = 0 3. The sum of the squares of deviations from the mean is the least
- 13. Median The median conveys the notion of being the middle most value with in the data distribution
- 14. Median For Un grouped data determined by first arranging the data in order of magnitude and then selecting the middle observation e.g. the median for the value 8, 10, 1, 3 and 5.
- 15. Median-Grouped data Media = Lm + N/2 – Cfbm cm fm Where Lm is the lower class boundary of the median class N is the total number of observations CFbm is the cumulative frequency of the class below the median class Cm is the class interval of the median class
- 16. Advantages/disadvantages of the median Advantages: Simple to calculate; It is representative of entire distribution; It is unique and representative of an actual figure in the distribution; Disadvantages: It cannot be subjected to further processing
- 17. Advantages/disadvantages of the median Advantages: Simple to calculate; It is representative of entire distribution; It is unique and representative of an actual figure in the distribution; Disadvantages: It cannot be subjected to further processing
- 18. Mode The Mode is the most common value in a given range of data Ungrouped data. In the following observations, 35, 42, 49, 49, 56, 70. the mode is 49
- 19. Mode Grouped data Lm + D1 x Cm D1 +D2 Where: Lm is the lower class boundary of the Model class D1 is the difference between the frequency of the modal class and frequency of the class before the modal class. D2 is the difference between the frequency of the modal class and frequency of the class after the modal class. Cm is the class width of the modal class
- 20. Advantages/disadvantages Advantages: It is simple Useful for qualitative data say the most handsome man; Disadvantages: Cannot be called unbiased Cannot be used to reconstruct the distribution Can not be further processed Some distributions are bimodal
- 21. Illustration Class frequency 400- 419 12 420- 439 27 440- 459 34 460- 479 15 480 499 8 Find The Mean, Mode and Median .
- 22. Exercise Class Frequency 36.0 – 37.7 6 37.8 – 39.5 7 39.6 – 41.3 24 41.4 – 43.1 7 43.2 – 44.9 2 45.0 – 46.7 4 Compute the mean, mode and median for the above distribution of distance to and from Work in Kilometers for UMI employees.