# Measures of location It is often useful to obtain summaries of certain aspects of the data. Most simple summary measurements can be divided into two types;

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Measures of location It is often useful to obtain summaries of certain aspects of the data. Most simple summary measurements can be divided into two types; firstly quantities which are typical of the data (with respect to frequency), and secondly, quantities which summaries the variability of the data. The former are known as measures of location and the latter as measures of spread. Suppose we have a sample of size n of quantitative data. We will denote the measurements by x1;x2; : : :xn. Central tendency is the middle point of a distribution. Measures of the central tendency is called as Measure of location. BIC Prepaid By:Rajyagor Bhargav

This is the most important and widely used measure of location. The sample mean of a set of data is =(x1+x2+….+xn)/ n =1/n n i=1 xi This is the location measure often used when talking about the average of a set of observations. Mean (Arithmetic mean) BIC Prepaid By:Rajyagor Bhargav

e.g. Generator (for power supply) 12345678910 Day out of service 7234821261394 Arithmetic mean = 7+23+4+8+2+12+6+13+9+4/10 =88/10 =8.8 days In this 10-months period, the generators were out of service for an average of 8.8 days. BIC Prepaid By:Rajyagor Bhargav

Conventional symbols A sample of population consist of n observations with a mean of The notation is different when we are computing measures for entire population. The mean of a population is symbolized by µ. The number of elements in population is denoted by N. BIC Prepaid By:Rajyagor Bhargav

Population arithmetic mean µ = x / N Where x = sum of all observation N = number of elements in the population. BIC Prepaid By:Rajyagor Bhargav

Sample arithmetic mean = x / n Where x = sum of all observation n = number of elements in the sample. BIC Prepaid By:Rajyagor Bhargav

Calculating the mean from ungrouped data. STUDENT S 1234567 Scor e 9776442 = x / n = 9+7+7+6+4+4+2 /7 =39/7 =5.6 points per students (sample mean) BIC Prepaid By:Rajyagor Bhargav

Calculating mean from Grouped Data Data are grouped by class. Each value of an observation falls some ware in one of the class. BIC Prepaid By:Rajyagor Bhargav

Sample arithmetic mean for grouped data. = (f X x ) / n Where = sample mean = symbol meaning the sum of f = frequency (number of observations) in each class x = midpoint for each class in the sample. n = number of total observation in the sample. BIC Prepaid By:Rajyagor Bhargav

We first calculate the midpoint of each class. Midpoint = Rounding of (starting of class + ending of class )/2 (f X x ) = multiply each midpoint by the frequency of observations in that class, sum all results, and Divide the sum by total number of observation in sample. BIC Prepaid By:Rajyagor Bhargav

e.g. Average Monthly salary of 600 customers of a particular bank. Class (Dollars)Frequency 0 – 49.9978 50.00 – 99.99123 100.00 – 149.99187 150.00 – 199.9982 200.00 – 249.9951 250.00 – 299.9947 300.00 – 349.0013 350.00 – 399.999 400.00 – 449.006 450.00 – 499.004 Total customers 600 BIC Prepaid By:Rajyagor Bhargav

To simplify our calculation of a mean from grouped data using a technique called coding. In this method we eliminate the problem of large or difficult midpoints. Instead of using actual midpoint to perform our calculation, we assign small value consecutive whole numbers called codes to each of the midpoints. Coding BIC Prepaid By:Rajyagor Bhargav

The integer zero can be assigned anywhere, but to keep integer small we will assign zero to midpoint in the middle of the frequency distribution. Then we can assign negative integer to values smaller then midpoint and positive integers to those larger. BIC Prepaid By:Rajyagor Bhargav

e.g. ClassMidpointCode 0 – 73.5-2 8 – 1511.5 16 – 2319.5 <- x 0 0 24 – 3127.51 32 – 3935.52 40 – 4743.53 BIC Prepaid By:Rajyagor Bhargav

Sample arithmetic mean of grouped data using code. = x 0 + w ((u X f) / n) Where = mean of sample x 0 = value of the midpoint assigned the code 0 w =numerical width of the lass interval. u = code assign to each class. f = frequency or number of observation in each class. n = total number observation in the sample. BIC Prepaid By:Rajyagor Bhargav

Advantages It represent a single number for whole data set. Its concept is familiar to all people and clear to all. every data set has a unique mean and easy to calculate. Mean is useful for performing statistical procedure like co0mparing mean from several data set. BIC Prepaid By:Rajyagor Bhargav

Disadvantages Reflect all the values in dataset. Although the mean is reliable in that it reflects all the values in the dataset, it may also affect by extreme value that are not representative of the rest of observation. MEMBER 1234567 TIMES IN MINUTE 4.24.34.74.85.05.19.0 BIC Prepaid By:Rajyagor Bhargav

Calculate the mean for above data = 4.2+4.3+4.7+4.8+5.0+5.1+9.0 / 7 = 37.1/7 =5.3 minutes ----- population mean If we compute mean for 6 members, and exclude 7 th. The answer for mean is 4.7. It would more appropriate to calculate mean without including such a extreme values. BIC Prepaid By:Rajyagor Bhargav

Use every data point in our calculation. We encounter with each and every data of our data set. So, it is more difficult with vast amount of data set. BIC Prepaid By:Rajyagor Bhargav

Unable to compute for open ended class. We are unable to compute the mean for a data set that has open-end class like >100 or <0 BIC Prepaid By:Rajyagor Bhargav

Weighted mean The weighted mean enables us to calculate an average that takes in to account the importance of each value to the overall total. w = (w X x)/ w where = symbol for the weighted mean w = weight assign to each observation. (w X x) = sum of the weight of each element times that with element w = sum of all weight BIC Prepaid By:Rajyagor Bhargav

Example LABUR INPUT IN MANUFACTURING COMPANY LABOUR HOURS PER UNIT OF OUTPUT Grade of Labor Hourly Wages(x) Product1Product2 Unskilled\$5.0014 Semiskilled\$7.0023 Skilled\$9.0053 BIC Prepaid By:Rajyagor Bhargav

A sample arithmetic mean of the labor rate. = x / n = \$5 + \$7 +\$9 /3 = \$21/3 = \$7.00 ---per hour Using this average rate, we would compute the labor cost of one unit of product 1 to be \$7(1+2+5) = \$56 and same for product2 will be \$70 which are incorrect. BIC Prepaid By:Rajyagor Bhargav

To solve this problem we have to consider weight for each observation. For product1 = (\$5 X 1) + (\$7 X 2) + (\$9 X 5) = \$64 / 8 = \$8 --per hour For product2 = (\$5 X 4) + (\$7 X 3) + (\$9 X 3) = \$68 / 10 = \$6.8 ---per hour BIC Prepaid By:Rajyagor Bhargav

W = (w x x) = (1/8 x \$5) + (2/8 x \$7) + (5/8 x \$9) 1/8 + 2/8 + 8/8 = \$8 / 1 = \$8.00 / hour BIC Prepaid By:Rajyagor Bhargav

do exercises A mini store advertises,if our average price are not equal or lower than everyone else's, you get it free. one of the customer came into the store one day and threw on the counter bills of sale of six items she bought from competitor for an average price less then that store. The item cost \$1.29\$2.97\$3.49\$5.00\$7.50\$10.95 Stores price for the same six are \$1.35\$2.89\$3.19\$4.98\$7.59\$11.50 Stores owner told the customer, my ad refers to a weighted average price of these items. our average is lower because our sales of these items have been:7912863 Is stores owner getting himself into or out of trouble by talking about weighted average? BIC Prepaid By:Rajyagor Bhargav

Geometric mean some times we dealing with quantities that changes over a period of time we need to know an average rate of change. Such as an average growth rate over a period of several year. In such cases, the simple arithmetic mean is inappropriate. we need to find the geometric mean simply called the G.M. BIC Prepaid By:Rajyagor Bhargav

How to calculate G.M. G.M. = n product of all x values E.g. Growth of \$100 deposit in a saving account YearInterest Rate Growth Factor Saving at End of year 1\$7\$1.07\$107.00 281.08115.56 3101.10127.12 4121.12142.37 5181.18168.00 BIC Prepaid By:Rajyagor Bhargav

Geometric mean 5 G.M. = 1.07 x 1.08 x 1.10 x 1.12 x 1.18 =5 1.679965 =1.1093 Average growth factor BIC Prepaid By:Rajyagor Bhargav

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