Measures of Central Tendency Mean,Median and Mode for Ungrouped Data Basic Statistics
Measures of Central Tendency In layman’s term, a measure of central tendency is an AVERAGE. It is a single number of value which can be considered typical in a set of data as a whole. For example, in a class of 40 students, the average height would be the typical height of the members of this class as a whole.
MEAN Among the three measures of central tendency, the mean is the most popular and widely used. It is sometimes called the arithmetic mean. If we compute the mean of the population, we call it the parametric or population mean, denoted by μ (read “mu”). If we get the mean of the sample, we call it the sample mean and it is denoted by (read “x bar”).
Mean for Ungrouped Data For ungrouped or raw data, the mean has the following formula. where = mean = sum of the measurements or values n = number of measurementsExample 1: Ms. Sulit collects the data on the ages of Mathematics teachers in Santa Rosa School, and her study yields the following: 38 35 28 36 35 33 40Solution: = 35Based on the computed mean, 38 is the average age of Mathematics teachers in SRS.
Your turn!Mang John is a meat vendor. The following are his sales forthe past six days. Compute his daily mean sales. Tuesday P 5 800 Wednesday 8 600 Thursday 6 500 Friday 4 300 Saturday 12 500 Sunday 13 400 Solution: = 51, 100 The average daily sales of Mang John is P51,100.
Weighted Mean Weighted mean is the mean of a set of values wherein each value or measurement has a different weight or degree of importance. The following is its formula: where = mean x = measurement or value w = number of measurements
ExampleBelow are Amaya’s subjects and the corresponding number of units and grades she got for the previous grading period. Compute her grade point average. Subject Units Grade Filipino .9 86 English 1.5 85 Mathematics 1.5 88 Science 1.8 87 Social Studies .9 86 TLE 1.2 83 MAPEH 1.2 87 = 86.1 Amaya’s average grade is 86.1
Your turn!James obtained the following grades in his five subjects forthe second grading period. Compute his grade point average. Subject Units Grade Math 1.5 90 English 1.5 86 Science 1.8 88 Filipino 0.9 87 MAKABAYAN 1.5 87 Solution: = 87.67 James general average is 87.67
Likert-type Question This is used if the researcher wants to know the feelings or opinions of the respondents regarding any topic or issues of interest.Next are examples of Likert-type statements. Respondents will choose the number which best represents their feeling regarding the statements. Note that the statements are grouped according to a theme. Choices 5 (SA) Strongly Agree 4 (A) Agree 3 (N) Neutral 2 (D) Disagree 1 (SD) Strongly Disagree
Students’ personal confidence in learning 5 4 3 2 1Statistics1. I am sure that I can learn Statistics2. I think I can handle difficult lessons inStatistics.3. I can get good grades in Statistics.Source: B.E. Blay, Elementary Statistics Below are the responses in the Likert-type ofstatements above. The table below shows the meanresponses and their interpretation. Using the formula forcomputing the weighted mean, check the correctness of thegiven means on the table. 5 4 3 2 1 Mean Interpretation 1 36 51 18 0 1 4.14 Agree 2 18 44 37 8 1 3.65 Agree 3 18 48 28 0 1 3.86 Agree
Your turn!Below is the result of the responses to the following Likert-type statements . Solve for the mean and give theinterpretation.Students’ perception on Statistics as a 5 4 3 2 1subject1. I think Statistics is a worthwhile, necessarysubject2. I will use Statistics in many ways as aprofessional3. I’ll need a good understanding of Statisticsfor my research work 5 4 3 2 1 Mean Interpretation 1 33 49 26 1 1 2 35 45 31 0 1 3 34 58 21 0 0
Properties of Mean1. Mean can be calculated for any set of numerical data, so it always exists.2. A set of numerical data has one and only one mean.3. Mean is the most reliable measure of central tendency since it takes into account every item in the set of data.4. It is greatly affected by extreme or deviant values (outliers)5. It is used only if the data are interval or ratio.
Your turn! Compute the median and interpret the result. 1. In a survey of small businesses in Tondo, 10 bakeries report the following numbers of employees: 15, 14, 12, 19, 13, 14 15, 18, 13, 19. 2. The random savings of 2nd year high school students reveal the following current balances in their bank accounts:Students A B C D E F G HCurrent Balances P340 350 450 500 360 760 800 740 3. The following are the lifetimes of 9 lightbulbs in thousands of hours. Lightbulb A B C D E F G H I Lifetime 1.1 1.1 1.2 1.1 1.4 .9 .2 1.2 1.7
Properties of Median1. Median is the score or class in the distribution wherein 50% of the score fall below it and another 50% lie.2. Median is not affected by extreme or deviant values.3. Median is appropriate to use when there are extreme or deviant values.4. Median is used when the data are ordinal.5. Median exists in both quantitative or qualitative data.
MODEExamples:Find the Mode.1. The ages of five students are: 17, 18, 23, 20, and 192. The following are the descriptive evaluations of 5 teachers: VS, S, VS, VS, O3. The grades of five students are : 4.0, 3.5, 4.0, 3.5, and 1.04. The weights of five boys in pounds are: 117, 218, 233, 120, and 117
Properties1. It is used when you want to find the value which occurs most often.2. It is a quick approximation of the average.3. It is an inspection average.4. It is the most unreliable among the three measures of central tendency because its value is undefined in some observations.
Your turn!Find the mode and interpret it.1. The following table shows the frequency of errors committed by 10 typists per minute. Typists A B C D E F G H I J No. of errors per min. 5 3 3 7 2 8 8 4 7 102. A random sample of 8 mango trees reveals the following number of fruits they yield Mango Tree A B C D E F G H No. of fruits 80 70 80 90 82 82 90 823. The following are the scores of 9 students in a Mathematics quiz.: 12, 15, 12, 8, 7, 15, 19, 24, 13