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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 measurements

Example 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      40
Solution:

             = 35
Based 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 for
the 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
Example
Below 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 for
the 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       1
Statistics
1. I am sure that I can learn Statistics
2. I think I can handle difficult lessons in
Statistics.
3. I can get good grades in Statistics.
Source: B.E. Blay, Elementary Statistics



       Below are the responses in the Likert-type of
statements above. The table below shows the mean
responses and their interpretation. Using the formula for
computing the weighted mean, check the correctness of the
given 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
Likert-type Mean Interpretation
 1.0 - 1.79   -   Strongly Disagree
 1.8 - 2.59   -   Disagree
 2.6 - 3.39   -   Neutral
 3.4 - 4.19   -   Agree
 4.2 - 5.00   -   Strongly Agree
Your turn!
Below is the result of the responses to the following Likert-
type statements . Solve for the mean and give the
interpretation.


Students’ perception on Statistics as a            5     4   3   2      1
subject
1. I think Statistics is a worthwhile, necessary
subject
2. I will use Statistics in many ways as a
professional
3. I’ll need a good understanding of Statistics
for 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 Mean
1. 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.
MEDIAN




16   17   18   19   20   21   22




16   17   18   19   20   21   22   23
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       H
Current 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 Median

1. 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.
MODE




Examples:
Find the Mode.
1. The ages of five students are: 17, 18, 23, 20, and 19
2. The following are the descriptive evaluations of 5
   teachers: VS, S, VS, VS, O
3. The grades of five students are : 4.0, 3.5, 4.0, 3.5, and
   1.0
4. The weights of five boys in pounds are: 117, 218, 233,
   120, and 117
Properties

1. 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   10

2. 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   82

3. The following are the scores of 9 students in a
   Mathematics quiz.: 12, 15, 12, 8, 7, 15, 19, 24, 13

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Measures of central tendency

  • 1. Measures of Central Tendency Mean,Median and Mode for Ungrouped Data Basic Statistics
  • 2. 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.
  • 3. 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”).
  • 4. 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 measurements Example 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 40 Solution: = 35 Based on the computed mean, 38 is the average age of Mathematics teachers in SRS.
  • 5. Your turn! Mang John is a meat vendor. The following are his sales for the 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.
  • 6. 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
  • 7. Example Below 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
  • 8. Your turn! James obtained the following grades in his five subjects for the 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
  • 9. 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
  • 10. Students’ personal confidence in learning 5 4 3 2 1 Statistics 1. I am sure that I can learn Statistics 2. I think I can handle difficult lessons in Statistics. 3. I can get good grades in Statistics. Source: B.E. Blay, Elementary Statistics Below are the responses in the Likert-type of statements above. The table below shows the mean responses and their interpretation. Using the formula for computing the weighted mean, check the correctness of the given 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
  • 11. Likert-type Mean Interpretation 1.0 - 1.79 - Strongly Disagree 1.8 - 2.59 - Disagree 2.6 - 3.39 - Neutral 3.4 - 4.19 - Agree 4.2 - 5.00 - Strongly Agree
  • 12. Your turn! Below is the result of the responses to the following Likert- type statements . Solve for the mean and give the interpretation. Students’ perception on Statistics as a 5 4 3 2 1 subject 1. I think Statistics is a worthwhile, necessary subject 2. I will use Statistics in many ways as a professional 3. I’ll need a good understanding of Statistics for 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
  • 13. Properties of Mean 1. 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.
  • 14. MEDIAN 16 17 18 19 20 21 22 16 17 18 19 20 21 22 23
  • 15. 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 H Current 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
  • 16. Properties of Median 1. 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.
  • 17. MODE Examples: Find the Mode. 1. The ages of five students are: 17, 18, 23, 20, and 19 2. The following are the descriptive evaluations of 5 teachers: VS, S, VS, VS, O 3. The grades of five students are : 4.0, 3.5, 4.0, 3.5, and 1.0 4. The weights of five boys in pounds are: 117, 218, 233, 120, and 117
  • 18. Properties 1. 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.
  • 19. 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 10 2. 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 82 3. The following are the scores of 9 students in a Mathematics quiz.: 12, 15, 12, 8, 7, 15, 19, 24, 13