Mathematics

Statistics

Session Objectives

Session Objectives Introduction Mean deviation from the mean Mean deviation from the median Variance and standard deviation Short-cut methods to find out the mean and standard deviation

Introduction We will study this topic based on your knowledge of the earlier classes. This includes knowledge of data representation and measures of central tendency — mean, median and mode. Mean of a continuous frequency distribution is given by

Introduction Median of a continuous frequency distribution is given by

Mean Deviation from the Mean Let us first understand what ‘mean deviation’ is. Mean deviation is the mean of the absolute deviations of a set of observations, taken from a definite central value (can be mean, median or anything else). The keyword to note in the above definition is ‘absolute’ — only the numerical value of the deviation is to be taken, ignoring the sign.

Mean Deviation from the Mean Mean deviation from the mean for raw data (unclassified) : In this case mean deviation from the mean for a set of n observations is given by Mean deviation from the mean for grouped data (classified) : In this case if xi’s are the mid-points of classes with frequency fi, then the mean deviation from the mean is given by

Mean Deviation from the Median The only difference here is that the mean is replaced by the value of the median. Mean deviation from the median for raw data (unclassified) In this case mean deviation from the median for a set of n observations is given by Mean deviation from the median for grouped data (classified) In this case if xi’s are the mid-points of classes with frequency fi , then the mean deviation from the median is given by

Variance and Standard Deviation The variance of a set of observation (xi) is the mean of the squares of deviations from mean of the observations . The variance is usually denoted by Var(X) or . If you now look at the definition above, there are 3 parts to it. So for a raw data of a set of n observations: (i) Deviations from mean of the observations (ii) Squares of deviations from mean (iii) Mean of the squares of deviations from mean

Variance and Standard Deviation Standard deviation is defined as the positive square root of the variance. The value of the variance and standard deviation for a grouped data is given by Variance, and Standard deviation (S.D.),

Short-cut Method to Find Out Mean and Variance ( ) In order to reduce the calculations involved in finding out the values of mean and variance for a grouped data, the following algorithm can be used to calculate the same. Algorithm for finding out the mean for a grouped data: Write down the frequency table with a column giving the class-marks (mid-points of class intervals) Choose a number ‘A’ (usually the middle or almost middle value of all xi’s) and take deviations di = xi– A about A. Divide each deviation by the class width h. Hence you get .

Short-cut Method to Find Out Mean and Variance ( ) Multiply the frequencies (fi) with the corresponding ui .Calculate the sum (fi ui ). Find the sum of all frequencies . Use the formula

Short-cut Method to Find Out Mean and Variance ( ) Similarly, we can also use a short-cut method to calculate the variance for a grouped data Write down the frequency table with a column giving the class-marks (mid-points of class intervals) Choose a number ‘A’ (usually the middle or almost middle value of all xi’s) and take deviations di = xi– A about A. Multiply the frequencies (fi) with the corresponding di. Calculate the sum (fi di ). Obtain the square of the deviations above (di2).

Short-cut Method to Find Out Mean and Variance ( ) Multiply the frequencies (fi) with the corresponding di2. Calculate the sum (fi di2). Find the sum of all frequencies . Use the formula

Class Test

Class Exercise - 1 The number of students absent in a school was recorded everyday for 147 days and the data is represented in the following frequency table. Obtain the median and describe what information it conveys. Also find the mean deviation from the median.

Solution Calculation of median and mean deviation Here, N = 147, The cumulative frequency just greater than is 140 and the value of x is 12. Hence, median = 12.

Solution contd.. The value of the median here signifies that for about half the number of days, approximately 12 students were absent. Mean deviation about median = 21.86

Class Exercise - 2 The following data represents the expenditure pattern of a student for the month of July. The student gets Rs. 50 everyday as a pocket money. Calculate the mean and standard deviation.

Solution Calculation of mean Hence, mean

Solution contd.. Calculation of standard deviation = 121.54 Hence, variance Hence, = 11.02

Class Exercise - 3 An absent-minded professor was computing certain experimental data to find the mean and standard deviation of 100 observations. He found mean to be 40 and the standard deviation to be 51. His assistant later found that the professor has, by mistake, read an observation value as 61, instead of the correct value of 91. Find the correct mean and standard deviation of the experimental data.

Solution Based on incorrect data, = 4030 Similarly, for standard deviation,

Solution contd... = 162601 Now, the correct value would be = 162601 + 4560 = 167161 = 1671.61 – 1602.41 = 69.2 So, the correct standard deviation,

Thank you