Measures of Central Tendency

Measures of Central Tendency

Measures of Central TendenciesNumbers which, in some sense, give the central or middle values of the data locates the center of the distribution of a set of data the most typical value of a set of data representative value of a given set of data

Mean arithmetic mean / average the sum of the values divided by the number of values which were added.

Mean of ungrouped data

Where - sample mean xi - ith observation/item in the sample n - number of observations in the sample

Example 1: find the mean of the sample: 7, 11, 11, 8, 12, 7, 6, 6

The sample mean is 8.5

Mean of ungrouped data

Example 2: find the mean of this sample: 18, 22, 25, 25, 26, 29, 45

Mean of ungrouped data

Weighted Mean

Where - weighted mean xi - ith observation/item in the sample wi weight of the ith observation

Examples of uses of weighted mean are in computing term GPA and in getting the mean responses for a Likert-type of questions.

It is used if the researcher wants to know the feelings or opinions of the respondents regarding any topic or issues of interest.Likert-type questions

Choices are:5 (SA) Strongly agree4 (A) Agree3 (N)Neutral2 (D)Disagree1 (SD)Strongly disagreeCheck appropriate box543211Student nurses serve as role models for their patients and the public. 2Student nurses should set a good example by not smoking.3Patient's chances of quitting smoking are increased if a student nurses advises him or her to quit.4Smoking is harmful to your health.5Smoking other tobacco products is harmful to a persons health.

Likert-type questions

54321Interpretation1711200291010032882041640005173000

Likert-type questions4.25Strongly Agree4.40Strongly Agree3.50Agree Likert-Type Mean Interpretation1.00 1.79 Strongly Disagree1.80 2.59 Disagree2.60 3.39 Neutral3.40 4.19 Agree4.20 5.00 Strongly AgreeStrongly Agree4.804.85Strongly Agree

Strongly Agree4.36

Mean for Grouped DataWhere f frequency of the class X Class Mark n sample size

ClassesFreq. (f)Class Mark(Xm)fXm 12 - 22 23 - 33 34 - 44 45 - 55 56 66476211728395061Total20

Mean for Grouped Data6819623410061659

Characteristics of the Mean1. It can be calculated for any set of numerical data, so it always exist.2. A set of numerical data has one and only one mean.4. It is greatly affected by extreme or deviant values.3. It is the most reliable since it takes into account every item in the set of data.

MedianThe median of a data is defined to be the middle value.

Note: it is important to arrange first the sample in ascending order before getting the median. Thus, when n is odd, the median is the center observation.When n is even, it is the average of the two center observations.

Example 3: find the median of the this sample: 7, 11, 11, 8, 12, 7, 6, 6

The sample median is 7.5

Solution: Arrange the observations in ascending order.6, 6, 7, 7, 8, 11, 11, 12

Since n = 8 (even), then

n/2 = 8/2 = 4 and (n/2) + 1=(8/2)+1=5 Thus,

Median

Example 4: find the median of this sample: 18, 22, 25, 25, 26, 29, 45

The sample median is 25Solution:The solution is already arranged in ascending order.

Since n = 7 (odd), then

(n + 1)/2 = (7 + 1)/2 = 4. Thus,= 25

Median

CharacteristicsThe score or class in a distribution, below which 50% of the score fall and above which another 50% lie.2. Not affected by extreme or deviant values.3. Appropriate to use when there are extreme or deviant values.

Illustration:18, 22, 25, 25, 26, 29, 45

Compare the mean (example 2) and the median (example 4)of the above sample.Mean = 27.1Median = 25 Which is the better measure of central tendency in thisexample? Why?

Median

1. It is used when we 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.Mode

ModeMode: VS2. The ages of 5 students are: 17, 18, 23, 20, & 19No Mode3. The grades of 5 students are: 4.0, 3.5, 4.0, 3.5, & 1.0Mode: 4.0 & 3.54. The weight of 5 persons in pounds are: 117, 218, 233, 120, & 117Mode: 1171. The following are the descriptive evaluation of 5 teachers. Teacher Evaluation A VS B S C VS D VS E S

Examples

ComparisonFactorMeanMedianModeType of dataQuantitativeQuantitativeQuantitative and QualitativeExtreme scoreproblemYesNoNoAlways measurableYesYesNoNumber of score110,1,2CharacteristicsAll scores included in computationMiddle valuePopular value