Measures of Central Tendency

Transcription

1 Measures of Central Tendency

2 Summary Measures Summary Measures Central Tendency Mean Median Mode Quartile Range Variance Variation Coefficient of Variation Standard Deviation

3 Measures of Central Tendency Central Tendency Average X n i 1 N i 1 n N X X i i Median Mode

4 The data in a sample is used to estimate the scores we expect to find in the entire population A number that describe a characteristic of a sample of scores is called a statistic, and the symbols for the different statistics are letters from the English alphabet. A number that describe a characteristic of a population of scores is called a parameter, and the symbols for the different parameter are letters from the Greek alphabet.

5 Central Tendency A measure of central tendency is a measure for the center point in the distribution of scores. A measure of central tendency is useful only if it represents accurately the distribution of scores on which it is based. Common measure of tendency: Mean, mode and median.

6 Mean (commonly referred as average ) The mean is arithmetic average for a population or sample. The symbol for the mean population is μ ( the Greek letter mu) The symbol for the mean of sample is X (X bar) _

7 Definitions: The arithmetic mean for a variable is computed by determining the sum of all the values of the variable in the data set, divided by the number of observations. The population arithmetic mean, μ (read mew ) is computed using all the individuals in a population. The population mean is a parameter The sample arithmetic mean, x (read xbar is computed using sample data. The sample mean is statistics.

8 Mean (Arithmetic Mean) Mean (arithmetic mean) of data values Sample mean X Population mean n X X X X i i N n X i i N Sample Size n Population Size X X X N n N

9 Mean (Arithmetic Mean) (continued) The most common measure of central tendency Affected by extreme values (outliers) Mean = 5 Mean = 6

10 Summation Notation Summation is represented by the capital Greek letter (sigma). X x1 x2 X variable. xi represents the ith observation. n number of observations. xn N The symbol Xi is read the summation of X sub i, where i goes from 1 to n. i=1

11 For example If we have X scores 26, 31, 27, 19, 43, and 37, then _ Σ X = = 183 N = 6 X = Σ X / N = 183/6 = 30.50

12 To calculate the mean for regular frequency distribution We sum all the scores and divide this sum by the number of scores in the data set _ X = Σ fx / N where Σ fx = F1x1 + f2x2 + +fnxn N = sum of frequencies, Σ fx

13 Advantage: Least sampling fluctuation compared to other two measure. Used in other statistical relationships Most representative measure of central tendency

14 Disadvantage: Not representative for skewed distribution

15 Median (Middle Number) The score below which 50 percent of the distribution falls is called the median (Md). Median is equal to the score associated with a relative cumulative of 50. Median is the score that is associated with the 50th percentile rank.

16 Median - Definition The median of a variable is the value that lies in the middle of the data when arranged in ascending order. That is, half the data are below and half the data are above the mean. We use Md to represent the median

17 Steps in Computing the Median of a Data Set. Step 1: Arrange the data in ascending order. Step 2: Determine the number of observations, N. Step 3: Determine the observation in the middle of the data set. If the number of observations is ODD, then the median is the data value that is exactly in the middle of the data set. That is, the median is the observation that lies in the position. If the number of observations is EVEN, then the median is the mean of the two middle observations in the data set. That is, the median is the mean of the data values that lie in the positions.

18 Median Robust measure of central tendency Not affected by extreme values Median = 5 Median = 5 In an ordered array, the median is the middle number If n or N is odd, the median is the middle number If n or N is even, the median is the average of the two middle numbers

19 To calculate median for grouped frequency For grouped frequency distribution we have a formula: Median = LRL + (50 LRLrcf) (URL LRL) (URLrcf LRLrcf) Where LRL = lower real limit of class interval containing median URL = upper real limit of class interval containing median LRLrcf = relative cumulative associated with lower real limit of class interval containing median URLrcf = relative cumulative associated with upper real limit of class interval containing median

20 Median = LRL + (50 LRLrcf) (URL LRL) (URLrcf LRLrcf)

21 To find median: 1. Find the class interval that contain the median : Apply the formula Median = (50-44)( ) (58-44) = (3) /14 = = 45.79

22 Median Advantage: Median is the most representative score for the data when distribution is skewed. It is not influenced by extreme scores in the distribution. Suitable for open-ended distribution. An openended distribution is one in which either the last class has no upper real limit, or the first class interval has no real limit. Thus, no mid-point for these open-ended class interval. Median is less subject to sample fluctuation

23 Median Disadvantage: Not sensitive to the value of scores in the distribution Like the mode, it is of little use in the statistical procedures. Median is primarily used for describing a set of data.

24 Mode The mode is the only indicator of central tendency that can be used with nominal data Can also be used with ordinal, interval, or ratio. Not a reliable measure of central tendency

25 Mode (Definition) The mode (Mo) of a variable is the most frequent observation of the variable that occurs in the data.

26 Computing the Mode: Tally the number of observations that occur for each data value. The data value that occurs most often is the mode. A data set can have no mode, one mode, or more than one mode. If there is no observation that occurs with the most frequency, we say the data has no mode.

27 Mode A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may may be no mode There may be several modes Mode = No Mode

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29 Example: Calculate mode for the following a) 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6, 4 b) 1, 2, 3, 4, 2, 5, 2, 3, 2, 4, 1, 4, 5, 1, 4 c) 1, 2, 3, 1, 4, 2, 3, 4, 1, 2, 4, 3

30 Q (a) :1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6, 4 x i frequency Mode = 4

31 Q (b) :1, 2, 3, 4, 2, 5, 2, 3, 2, 4, 1, 4, 5, 1, 4 x i frequency There are 2 score that has the highest frequency, 2 and 4. Mode = 2 and 4 ( Bimodal)

32 Example: Find the mode

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34 Modal class interval is the class interval that has the largest frequency. The above example, the modal class interval is 39-41, which has a midpoint of 40. Thus the mode for this grouped frequency of 40

35 To calculate mode: (See example mod data terkumpul)

36 Mode Advantage using mode: Easy to compute Mode is the preferred measure of central tendency for nominal scales

37 Mode Disadvantage using mode: There are some distributions with more than one mode, i.e bimodal and multimodal distributions. In these cases here is no single value of the mode use as the measure of central tendency. Mode is subject to a substantial amount of sampling fluctuation. Sampling fluctuation refers to the difference between the sample statistics from sample of the same size which have been drawn from the same population. Mode is only used for descriptive purposes; Often used when researcher wants to summarize nominal data

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39 Copyright 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning Figure 3.11 Central tendency and skewed distributions

40 Copyright 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning Figure 3.10 Central tendency and symmetrical distributions

41 Mean/Median vs skewness

42 Mean/Median vs skewness

43 Mean/Median vs skewness

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45 Relative merits of the Mean, Median, and Mode

46 Mean is the most widely used measure of central tendency. Takes all measurements into consideration. Easy to manipulate with algebra. Preferred measure of central tendency for symmetric distributions.

47 Median is resistant to extreme values. For data sets with unusually large or small values relative to the entire data set or when the data are skewed, the median is the preferred measure of central tendency. Examples of the use of the median include income levels and housing prices, which are skewed right, and life span (age), which skewed left. In economic statistics, it is oftentimes desirable to disregard extreme variates which may be due to unusual circumstances.

48 Mode is resistant to extreme values. Can be ambiguous (tdk tentu)if there are two modes that are not adjacent

49 Quartiles Split Ordered Data into 4 Quarters 25% 25% 25% 25% Position of i-th Quartile Data in Ordered Array: Position of Q1 2.5 Q Q 1 Q 3 Q 1 Q Q 2 3 Q and Are Measures of Noncentral Location Q 2 = Median, A Measure of Central Tendency i i n 1 4

50 Thank You