Session 1.6 Measures of Central Tendency

Transcription

1 Session 1.6 Measures of Central Tendency

2 Measures of location (Indices of central tendency) These indices locate the center of the frequency distribution curve. The mode, median, and mean are three indices of central tendency: Mode = the most frequently occurring observation Median = that measurement level below which half the observations fall, the 5th percentile Mean = sum of the observed measurements Number of observations For any symmetrical distribution, the mean, median, and mode will be identical.

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6 Common measures of central tendency The arithmetic mean The median The mode

7 The Arithmetic Mean The balancing point of the distribution curve (even if not strictly normal). he sum of negative deviations from the mean exactly equals he sum of positive deviations from the mean.

8 Calculation x = n x i

9 Example x 1- To calculate the numerator, sum the individual observations: x I = = 168 i n 2- For the denominator, count the number of observations: n = 6 3- To calculate the mean: x 168 = = days a n o u t b r e

10 Advantages & Disadvantages of the Mean Advantages: -Mathematical center of a distribution. -Good for interval and ratio data. -Does not ignore any information. -Inferential statistics is based on mathematical properties of the mean. Disadvantages: -Influenced by extreme scores and skewed distributions. -May not exist in the data.

11 The Median The middle of a set of data that has been put into ascending or descending order. It is the value that divides a set of data into two halves, with one half of the observations being larger than the median value, and one half smaller.

12 Calculation 1- Arrange the observations in increasing or decreasing order. 2- Find the middle rank with the following formula: Middle rank = (n + 1) 2 a- If n is odd, the middle rank falls on an observation. b- If n is even, the middle rank falls between two observation.

13 Calculation (cont.) 3- Identify the value of median: a- If the middle rank falls on a specific observation (n is odd), the median is equal to the value of that observation. b- If the middle rank falls between two observations (n is even), the median is equal to the average of the values of these observations.

14 Example with an odd n Find the median for the following set of data: 13, 7, 9, 15, Arrange the observations. e.g. 7, 9, 11, 13, 15 or 15, 13, 11, 9, 7 2- Find the middle rank: (n + 1) (5 + 1) Middle ran k = = = Identify the value of the median. It is 11

15 Example with an even n Find the median for the following set of data: 15, 7, 13, 9, 10, Arrange the observations. e.g. 7, 9, 10, 11, 13, Find the middle rank: (n + 1) (6 + 1) Middle rank = = = Identify the value of the median. It is equal to the average of the values of the third and fourth observations, so the median is ( ) Median = =

16 Advantages & Disadvantages of the Media Advantages: - Not influenced by extreme scores or skewed distribution. - Good with ordinal data. -Easier to compute than the mean. -Considered as the typical observation. Disadvantages: - May not exist in the data. - Does not take actual values into account.

17 The Mode The value that occurs most often in a set of data. There may be more than one mode for a distribution of data.

18 Calculation We usually find the mode by creating a frequency distribution in which we count how often each value occurs. If we find that every value occurs only once, the distribution has no mode. If we find that two or more values are tied as the most common, the distribution has more than one mode.

19 Example Find the mode for the following set of data: 29, 31, 24, 29, 30 and 25 days 1- Arrange the data into a frequency distribution: xi Identify the value that occurs most often Mode = 29 days f i

20 Advantages & Disadvantages of the Mode Advantages: - Good with nominal data. - Bimodal distribution might verify clinical observations (pre and post-menopausal breast cancer). - Easy to compute and understand. - The score exists in the data set. Disadvantages: - Ignore most of the information in a distribution. - Small samples may not have a mode - More than one mode might exist.

21 Central tendency and skew Since the mean is drawn in the direction of outliers. And median is unaffected by number of outliers, The mode is always the most frequent observation. Relationship between mean, median and mode may give some indicating of the shape of the frequency distribution without having to create a histogram

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25 Skewed right (positive) Skewed left (negative) (+) (-) Mode Median Mean Mean Median Mode