Training Course on Basic Statistics for Research August 24-28, 2009 STATISTICAL RESEARCH AND TRAINING CENTER J and S Building, 104 Kalayaan Avenue, Diliman, Quezon City Measures of Central Tendency Prepared by: Josefina V. Almeda Professor and College Secretary School of Statistics University of the Philippines, Diliman August 2009

2 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 Measures of Central Tendency OUTLINE  Mean  Median  Mode

3 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 Central Tendency Mean Median Mode Other Locations Summary Measures Variation Variance Standard Deviation Coefficient of Variation Range Quartiles Describing Data with Summary Measures

4 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 Measures of Central Tendency  Measure of central tendency is an index of the central location of a distribution. It is a single value that is used to identify the “center” of the data or the typical value.  Precise yet simple  Most representative value of the data

5 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 The arithmetic mean is the sum of all observed values divided by the total number of observations. The population mean for a finite population with N elements, denoted by the Greek letter (lowercase Greek letter mu), is The Arithmetic Mean

6 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 The sample mean for a finite sample with n elements, denoted by The population mean is a parameter while the sample mean is a statistic.

7 Statistical Research and Training Center Training Course on Basic Statistics for Research August , Given the number of children of a sample of 10 currently married women: 3, 4, 2, 5, 1, 3, 4, 2, 3, 3, find the mean number of children of the currently married women. Solution: We compute for the sample mean. The mean number of children of currently married women is 3. Examples of Arithmetic Mean

8 Statistical Research and Training Center Training Course on Basic Statistics for Research August , Given the incidence of alleged human rights violations by region for the year 2004, find the mean incidence of alleged human rights violations. NCR 133 CAR 11 Region1 2 Region 2 16 Region 3 41 Region 4 57 Region 5 30 Region 6 49 Region 7 44

9 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 Region 8 71 Region 9 73 Region Region Region Region Solution: We get the population mean incidence of alleged human rights violations The mean incidence of alleged human rights violations per region is 59.3.

10 Statistical Research and Training Center Training Course on Basic Statistics for Research August , The mean is the most common measure of central tendency since it employs every observed value in the calculation. 2. It may or may not be an actual observed value in the data set. 3.We may compute the mean for both ungrouped and grouped data sets. 4. Extreme observations affect the value of the mean especially if the number of observations is small. Properties of the Mean

11 Statistical Research and Training Center Training Course on Basic Statistics for Research August , The value of the mean always exists and unique. 6.It is a widely understood measure of central tendency. 7.We use the mean if the distribution is not so asymmetrical; when we give equal importance to the effect of all observed values; and when we compute other statistics later on.

12 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 * if the individual values do not have equal importance, then we compute for the weighted mean. * We assign weights to the observed values of the data set before we can get the weighted mean. The Weighted Mean

13 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 If we assign a weight to each observation where i = 1, 2,…, n, and n is the number of observations in the sample, then the weighted sample mean is given by Formula of Weighted Mean

14 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 Suppose a government agency gives scholarship grants to employees taking graduate studies. Courses in graduate studies earn credits of 1, 2, 3, 4, or 5 units. They can get a partial scholarship for the next semester if they get a weighted average of 1.5 to 1.75 and a full scholarship if the average is better than 1.5, which means an average of 1.0 to What kind of scholarship will the 2 employees get given their grades for the previous semester? Example of Weighted Mean

15 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 Employee A Employee B Subjects UnitsGradeSubjects UnitsGrade A11.0A12.0 B21.25B21.75 C31.5C3 D41.75D41.25 E52.0E51.0 Consider the grades of the two employees in the previous semester:

16 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 Solution: We let the units be the weights Wi and the grade is the Xi. Weighted average of employee A: Weighted average of employee B: Thus, employee A will get a partial scholarship while employee B will get a full scholarship.

17 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 We can obtain the mean of several data sets given the means and number of observations of each data set. This is what we call the combined mean. Suppose that k finite populations having measurements, respectively, have means The combined population mean, of all the populations is The Combined Population Mean

18 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 If random samples of size, selected from these k populations, have the means respectively, the combined sample mean of all the sample data is

19 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 = = Thus, Example of the Combined Mean The Philippines have 6028 male children deaths and 4948 female children deaths for the age group 1-4 in The average number of deaths for male and female children is and What is the combined population mean for both sexes? = = Thus, The average number of deaths for children 1-4 years old for both sexes is 346. Solution: We let = 6028 and N2 = 4948.

20 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 * It divides an ordered observation into two equal parts so that half of the observations are below its value and the other half are above its value. * It is the positional middle of the array. Example: If the median annual family income of 500 families is P185,000, then this implies that half of the 500 families (250 families) have annual family income lower than P185,000 and the other half (250 families) have annual family income higher than P185,000. The Median

21 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 * The first step in finding the median, denoted by Md, is to arrange the observations in an array. Case 1: If the number of observations n is odd, the median is the middle observed value in the array. Computation of the Median Case 2: If the number of observations n is even, the median is the average of the two middle observed values in the array.

22 Statistical Research and Training Center Training Course on Basic Statistics for Research August , The annual per capita poverty threshold in pesos of the different regions of the Philippines are as follows: 15,693, 13,066, 12,685, 11,128 13,760, 13,657, 11,995, 11,372, 11,313, 9,656, 9,518, 9,116, 10,503, 10,264, 10,466, 10,896, 12,192. Solution: We arrange the 17 annual per capita poverty threshold in pesos of the 17 regions of the Philippines from lowest to highest. Examples of the Median

23 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 Since n = 17 is odd, the median is the middle observed value in the array. That is the median is P11, Interpretation: Half of the 17 regions have annual per capita poverty threshold of P11,313 and the other half have annual per capita poverty threshold higher than P11,313 pesos. Array: 9116, 9518, 9656, 10264, 10466, 10503, 10896, 11128, 11313, 11372, 11995, 12192, 12,685, 13066, 13657, 13760, 15693

24 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 Array: 33315, 35945, 42860, 82616, 94079, , , , , , , , , , , n = 16 is even Interpretation: 50% of the 16 regions have number of telephone lines less than and the upper 50% have number of telephone lines more than The following are the number of telephone lines of 16 regions for the year 2004: , 94079, , 42860, , , , , , , 35945, , , 82616, , Find the median.

25 Statistical Research and Training Center Training Course on Basic Statistics for Research August , The median is a positional measure. This implies that extreme values affect the median less than the mean. 2.We use the median as a measure of central tendency if we wish the exact middle value of the distribution, when there are extreme observed values, and when the frequency distribution table has open-ended class intervals. Characteristics of the Median

26 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 * is the observed value that occurs with the greatest frequency in a data set. * determine the mode by counting the frequency of each observed value and finding the observed value with the highest frequency of occurrence. * Generally, the mode is a less popular measure of central tendency as compared to the mean and the median. The Mode

27 Statistical Research and Training Center Training Course on Basic Statistics for Research August , Given the data on number of children of 12 currently married women: 2, 2, 1, 1, 1, 3, 3, 4, 4, 2, 2, 2. Find the mode. By inspection, the mode is 2. Interpretation: The most frequent number of children among the 12 currently married women is 2. 2.Given the data on number of cases resolved by a 10 lawyers: 5, 4, 1, 1, 3, 3, 2, 1, 3, 0. Find the mode. The modes are 1 and 3. Examples of Mode

28 Statistical Research and Training Center Training Course on Basic Statistics for Research August , Given the data on number of cases handled by 14 PAO lawyers : 629, 645, 356, 656, 231, 455, 412, 289, 444, 452, 642, 225, 335, 411. Find the mode.

29 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 Characteristics of the Mode 1.The mode gives the most typical value of a set of observations. 2.Few low or high values do not easily affect the mode. 3.The mode is sometimes not unique and does not exist. 4.We can have several modes for one data set. If there is one mode, it is unimodal. If there are two modes, we call it bimodal. If there are more than two modes, then we call it multimodal. 5.The value of the mode is always one of the observed values in the data set. 6.We can get the mode for both quantitative and qualitative types of data.

30 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 Given the number of cellular mobile telephone subscribers for the year 2001, what is the mode? Telephone OperatorNumber of Subscribers EXTELCOM194,452 GLOBE TELECOM 5,405,415 ISLACOM181,614 PILTEL 1,483,838 SMART 4,893,844 Example of Mode for Qualitative Data

31 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 * In performing calculations, we only round-off the final answer and not the transitional values. * The final answer should increase by one digit of the original observations. Example: The mean of the data set 3, 4, and 6 is ….. Round this figure to the nearest tenth since the original observed values are whole numbers. Thus, the mean becomes 4.3. Example: If the original observed values have one decimal place like 4.5, 6.3, 7.7, 8.9, then we round the final answer to two decimal places. Thus, if we get the mean, the final answer is Round-Off Rule

Training Course on Basic Statistics for Research August 24-28, 2009 STATISTICAL RESEARCH AND TRAINING CENTER J and S Building, 104 Kalayaan Avenue, Diliman, Quezon City Thank you.