Statistics Chapter 3Statistics Chapter 3Measures of Central TendencyMeasures of Central Tendency

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Norhatta MohamadNorhatta Mohamad

Agenda

Objective

Introduction to Measures of Central Tendency

Mean Discussion

Median Discussion

Mode Discussion

Skewness Discussion

Test your understanding

The End

Objective

At the end of this chapter, you should be able to

1. Construct a frequency distribution table

2. Calculate mean, median and mode

3. Obtain median from cumulative frequency curve

4. Obtain mode from a histogram

5. Draw a skewed to the right graph and a skewed to the left graph with the correct position of mean, median and mode

Introduction to Measure of Central Tendency

There are three type of measures of central tendency

Mean

The mean or the arithmetic mean, is the most frequently used measure of central tendency.

is usually used as the symbol for

sample mean while is the symbol for population mean.

Mean for Ungrouped Data

Example1. Find the mean for 4, 6, 1, 8, 6, 3, 7 and 6

Example … Continue .

4. Find the value of M if the mean for M, 4.6, 2.7, 5.1 and 4.7 is 3.86.

Mean for grouped data

Q&A

End of MEAN discussion

Median

Definition:

Median is the _______________ in a data set that has been arranged in either increasing or decreasing order. It divides an ordered data set into two equal parts.

Median for Ungrouped Data

The steps are

1.Arrange the given data set in increasing (or decreasing) order.

2.Find the middle position at th position, where n is the number of data:

If the number of observations in a data set is odd, then the median is the middle term of the ordered set.

If the number of observations in a data set is even, then the median is by the average of the two middle terms.

2

1n

Example 1Find the median for the following sets of data:

a.

b.

Example 2

Example 3

Find a value of M if the median for

a. M, 1, 4, 2, 4, 7, 1 is 3.

b. M, 1, 4, 2, 3, 4, 4, 7 is 3.5. M is an integer.

Median for grouped data

Example 4

Q&A

End of MEDIAN discussion

Mode

Definition:

Mode is the data that appears most _______, that is the data with highest frequency.

If two data have the highest number of frequency, then there are two modes and they are called __________.

If there are three data with highest frequency, then there are __________, etc.

If the frequency of all data is equal, then the set of data has __________.

Mode for Ungrouped Data

Example 1

Solution :

Mode for Ungrouped Data

Example 2

Solution :

Mode for grouped data

Example 3

Solution :

Solution :

Q&A

End of MODE discussion

SkewnessSkewness describes the asymmetry of a distribution. A skewed distribution therefore has one tail longer than the other. A _______________ distribution has a longer tail to the right:A _________________ distribution has a longer tail to the left:A distribution with no skew (e.g. a normal distribution) is symmetrical: In a perfectly symmetrical, non-skewed, distribution the mean, median and mode are equal. As distributions become more skewed the difference between these different measures of _____________gets larger.

Q&A

End of Skewness discussion

Test Your Understanding 1

The following are donations collected for a local charity:

1. Arrange the scores from highest to lowest.

2. Find the MODE (modal donation). 3. Find the MEDIAN (median donation). 4. Determine the sum of the scores. 5. Find the MEAN (mean or average

donation).

Answers :

Test Your Understanding 2

1. Three commonly used measures of ______ are the mean, the mode, and the ______.

2.Usually the scores will cluster around the most frequently occurring score, the ______.

3.The midpoint that separates the distribution into equal-sized halves is the ______.

4.The most stable, or reliable measure of central tendency tends to be the ______.

5.The ________is sensitive to the value of every score in the distribution; this is not true of the ______ or the ______.

Test Your Understanding 2

6.Distributions that are symmetrical and have a specified bell shape are termed ______ distributions.

7.In a true ______ distribution the mode, median, and mean have the same ______.

8.A test may be so difficult that there are many ______ scores and few ______ scores.

9.Such a distribution would be described as ______ skewed.

10. In severely skewed data sets, the ______ may be the most accurate measure of central tendency.

Test Your Understanding 2

11. On a very easy test, the ______ will be greater than the ______ but less than the ______.

12. If the mean IQ for a group of MIITs is found to be 110 and the median IQ is 100, the distribution is probably ______ skewed.

Solution:

1. Three commonly used measures of __________ are the mean, the mode, and the ____________.

2. Usually the scores will cluster around the most frequently occurring score, the __________.

3. The midpoint that separates the distribution into equal-sized halves is the ____________

4. The most stable, or reliable measure of central tendency tends to be the __________

5. The __________ is sensitive to the value of every score in the distribution; this is not true of the _______ or the _______

6. Distributions that are symmetrical and have a specified bell shape are termed _________ distributions.

7. In a true __________ distribution the mode, median, and mean have the same _______

8. A test may be so difficult that there are many _____ scores and few _______ scores.

Solution:

9. Such a distribution would be described as _________ skewed.

10. In severely skewed data sets, the ________ may be the most accurate measure of central tendency.

11. On a very easy test, the ________ will be greater than the ______ but less than the _______

12. If the mean IQ for a group of MIITs is found to be 110 and the median IQ is 100, the distribution is probably __________ skewed.

Test Your Understanding 3

Listed below are scores from a test of human relations skills that was taken by 30 managers in a major industrial firm. The test consisted of 40 multiple choice questions, so it was possible to score as low as 0 or as high as 40.

Organize these raw scores into a grouped frequency distribution consisting of 7 class intervals (first class 17 – 19). Calculate the median by calculation method.

Solution:

Thank You