Lecture Notes on Basic Statistics - Measures of Central Tendency PDF

Lecture notes on Basic Statistics Chapter 3: MEASURES OF CENTERAL TENDENCY 6CHAPTER 3 3. MEASURES OF CENTERAL TENDENCY 3.1 Introduction  When we want to make comparison between groups of numbers it is good to ha...

Lecture notes on Basic Statistics Chapter 3: MEASURES OF CENTERAL TENDENCY 6CHAPTER 3 3. MEASURES OF CENTERAL TENDENCY 3.1 Introduction  When we want to make comparison between groups of numbers it is good to have a single value that is considered to be a good representative of each group. This single value is called the average of the group. Averages are also called measures of central tendency.  An average which is representative is called typical average and an average which is not representative and has only a theoretical value is called a descriptive average. A typical average should posses the following:  It should be rigidly defined.  It should be based on all observation under investigation.  It should be as little as affected by extreme observations.  It should be capable of further algebraic treatment.  It should be as little as affected by fluctuations of sampling.  It should be ease to calculate and simple to understand. 3.2 Objectives:  To comprehend the data easily.  To facilitate comparison.  To make further statistical analysis. 3.3 The Summation Notation:  Let X1, X2 ,X3 …XN be a number of measurements where N is the total number of observation and Xi is ith observation.  Very often in statistics an algebraic expression of the form X1+X2+X3+...+XN is used in a formula to compute a statistic. It is tedious to write an expression like this very often, so mathematicians have developed a shorthand notation to represent a sum of scores, called the summation notation.  The symbol is a mathematical shorthand for X1+X2+X3+...+XN The expression is read, "the sum of X sub i from i equals 1 to N." It means "add up all the numbers." Example: Suppose the following were scores made on the first homework assignment for five students in the class: 5, 7, 7, 6, and 8. In this example set of five numbers, where N=5, the summation could be written: Page 1 Lecture notes on Basic Statistics Chapter 3: MEASURES OF CENTERAL TENDENCY The "i=1" in the bottom of the summation notation tells where to begin the sequence of summation. If the expression were written with "i=3", the summation would start with the third number in the set. For example: In the example set of numbers, this would give the following result: The "N" in the upper part of the summation notation tells where to end the sequence of summation. If there were only three scores then the summation and example would be: Sometimes if the summation notation is used in an expression and the expression must be written a number of times, as in a proof, then a shorthand notation for the shorthand notation is employed. When the summation sign "∑" is used without additional notation, then "i=1" and "N" are assumed. For example: PROPERTIES OF SUMMATION 1. where k is any constant 2. where k is any constant 3. where a and b are any constant Page 2 Lecture notes on Basic Statistics Chapter 3: MEASURES OF CENTERAL TENDENCY 4. The sum of the product of the two variables could be written: Example: considering the following data determine X Y 5 6 7 7 7 8 6 7 8 8 a) e) b) f) c) g) d) h) Solutions: a) b) c) Page 3 Lecture notes on Basic Statistics Chapter 3: MEASURES OF CENTERAL TENDENCY d) e) f) g) h) 3.4 Types of measures of central tendency There are several different measures of central tendency; each has its advantage and disadvantage.  The Mean (Arithmetic, Geometric and Harmonic)  The Mode  The Median  Quantiles (Quartiles, Deciles and Percentiles) The choice of these averages depends up on which best fit the property under discussion. a) The Arithmetic Mean I) For Ungrouped Data  Is defined as the sum of the magnitude of the items divided by the number of items.  The mean of X1, X2 ,X3 …Xn is denoted by A.M ,m or and is given by:  If X1 occurs f1 times, if X2occurs f2 times, … , if Xn occurs fn times Page 4 Lecture notes on Basic Statistics Chapter 3: MEASURES OF CENTERAL TENDENCY Then the mean will be , where k is the number of classes and Example: Obtain the mean of the following number 2, 7, 8, 2, 7, 3, 7 Solution: Xi fi Xifi 2 2 4 3 1 3 7 3 21 8 1 8 Total 7 36 II) For Grouped Data If data are given in the shape of a continuous frequency distribution, then the mean is obtained as follows: Xi =the class mark of the ith class and fi = the frequency of the ith class Example: calculate the mean for the following age distribution. Class frequency 6- 10 35 11- 15 23 16- 20 15 21- 25 12 26- 30 9 31- 35 6 Page 5 Lecture notes on Basic Statistics Chapter 3: MEASURES OF CENTERAL TENDENCY Solutions:  First find the class marks  Find the product of frequency and class marks  Find mean using the formula. Class fi Xi Xifi 6- 10 35 8 280 11- 15 23 13 299 16- 20 15 18 270 21- 25 12 23 276 26- 30 9 28 252 31- 35 6 33 198 Total 100 1575 Exercises: 1. Marks of 75 students are summarized in the following frequency distribution: Marks No. of students 40-44 7 45-49 10 50-54 22 If 20% of 55-59 f4 the students have marks between 55 and 59 60-64 f5 i. 65-69 6 Find the missing frequencies f4 and f5. ii. 70-74 3 Find the mean. III) Special properties 1. The sum of the deviations of a set of items from their mean is always zero. i.e. 2. The sum of the squared deviations of a set of items from their mean is the minimum. i.e. 3. If is the mean of observations, if is the mean of observations, … , if is the mean of observation, then the mean of all the observation in all groups often called the combined mean is given by: Page 6 Lecture notes on Basic Statistics Chapter 3: MEASURES OF CENTERAL TENDENCY Example: In a class there are 30 females and 70 males. If females averaged 60 in an examination and boys averaged 72, find the mean for the entire class. Solutions: 4. If a wrong figure has been used when calculating the mean the correct mean can be obtained with out repeating the whole process using: Where n is total number of observations. Example: An average weight of 10 students was calculated to be 65.Latter it was discovered that one weight was misread as 40 instead of 80 kg. Calculate the correct average weight. Solutions: 5. The effect of transforming original series on the mean. a) If a constant k is added/ subtracted to/from every observation then the new mean will be the old mean± k respectively. b) If every observations are multiplied by a constant k then the new mean will be k*old mean Page 7 Lecture notes on Basic Statistics Chapter 3: MEASURES OF CENTERAL TENDENCY Example: 1. The mean of n Tetracycline Capsules X1, X2, …, Xn are known to be 12 gm. New set of capsules of another drug are obtained by the linear transformation Yi = 2Xi – 0.5 ( i = 1, 2, …, n ) then what will be the mean of the new set of capsules Solutions: 2. The mean of a set of numbers is 500. a) If 10 is added to each of the numbers in the set, then what will be the mean of the new set? b) If each of the numbers in the set are multiplied by -5, then what will be the mean of the new set? Solutions: IV) Weighted Mean  When a proper importance is desired to be given to different data a weighted mean is appropriate.  Weights are assigned to each item in proportion to its relative importance.  Let X1, X2, …Xn be the value of items of a series and W 1, W2, …Wn their corresponding weights , then the weighted mean denoted is defined as: Example: A student obtained the following percentage in an examination: English 60, Biology 75, Mathematics 63, Physics 59, and chemistry 55.Find the students weighted arithmetic mean if weights 1, 2, 1, 3, 3 respectively are allotted to the subjects. Solutions: V) Merits:  It is based on all observation. Page 8 Lecture notes on Basic Statistics Chapter 3: MEASURES OF CENTERAL TENDENCY  It is suitable for further mathematical treatment.  It is stable average, i.e. it is not affected by fluctuations of sampling to some extent.  It is easy to calculate and simple to understand. Demerits:  It is affected by extreme observations.  It can not be used in the case of open end classes.  It can not be determined by the method of inspection.  It can not be used when dealing with qualitative characteristics, such as intelligence, honesty, beauty. b) The Geometric Mean  The geometric mean of a set of n observation is the nth root of their product.  The geometric mean of X1, X2 ,X3 …Xn is denoted by G.M and given by:  Taking the logarithms of both sides The logarithm of the G.M of a set of observation is the arithmetic mean of their logarithm. Example: Find the G.M of the numbers 2, 4, 8. Solutions: Remark: The Geometric Mean is useful and appropriate for finding averages of ratios. c) The Harmonic Mean I) Simple Harmonic Mean The harmonic mean of X1, X2 , X3 …Xn is denoted by H.M and given by: Page 9 Lecture notes on Basic Statistics Chapter 3: MEASURES OF CENTERAL TENDENCY , This is called simple harmonic mean. II) Harmonic Mean With FD In a case of frequency distribution: , If observations X1, X2, …Xn have weights W1, W2, …Wn respectively, then their harmonic mean is given by , This is called Weighted Harmonic Mean. Remark: The Harmonic Mean is useful and appropriate in finding average speeds and average rates. Example: A cyclist pedals from his house to his college at speed of 10 km/hr and back from the college to his house at 15 km/hr. Find the average speed. Solution: Here the distance is constant The simple H.M is appropriate for this problem. X1= 10km/hr X2=15km/hr 2) The Mode I) Mode for Ungrouped (Discrete) FD - Mode is a value which occurs most frequently in a set of values - The mode may not exist and even if it does exist, it may not be unique. Page 10 Lecture notes on Basic Statistics Chapter 3: MEASURES OF CENTERAL TENDENCY - In case of discrete distribution the value having the maximum frequency is the modal value. Examples: 1. Find the mode of 5, 3, 5, 8, 9 Mode =5 2. Find the mode of 8, 9, 9, 7, 8, 2, and 5. It is a bimodal Data: 8 and 9 3. Find the mode of 4, 12, 3, 6, and 7. No mode for this data. - The mode of a set of numbers X1, X2, …Xn is usually denoted by. II) Mode for Grouped data If data are given in the shape of continuous frequency distribution, the mode is defined as: Where: Note: The modal class is a class with the highest frequency. Example: Following is the distribution of the size of certain farms selected at random from a district. Calculate the mode of the distribution. Size of farms No. of farms 5-15 8 15-25 12 25-35 17 Page 11 Lecture notes on Basic Statistics Chapter 3: MEASURES OF CENTERAL TENDENCY 35-45 29 45-55 31 55-65 5 65-75 3 Solutions: III) Merits:  It is not affected by extreme observations.  Easy to calculate and simple to understand.  It can be calculated for distribution with open end class IV) Demerits:  It is not rigidly defined.  It is not based on all observations  It is not suitable for further mathematical treatment.  It is not stable average, i.e. it is affected by fluctuations of sampling to some extent.  Often its value is not unique. Note: being the point of maximum density, mode is especially useful in finding the most popular size in studies relating to marketing, trade, business, and industry. It is the appropriate average to be used to find the ideal size. Page 12 Lecture notes on Basic Statistics Chapter 3: MEASURES OF CENTERAL TENDENCY 3) The Median - In a distribution, median is the value of the variable which divides it in to two equal halves. - In an ordered series of data median is an observation lying exactly in the middle of the series. It is the middle most value in the sense that the number of values less than the median is equal to the number of values greater than it. -If X1, X2, …Xn be the observations, then the numbers arranged in ascending order will be X, X, …X[n], where X[i] is ith smallest value. X< X< …