/ / / x means sum of scores and n =/ f is the number of scores. J 14. Data. Knowing More. Mean, Median, Mode
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1 Mean, Median, Mode The mean of a data set is written as xr (pronounced x-bar ). It is the arithmetic average of the data set. sumofscores x x x r = or xr = = number of scores n f where x means sum of scores and n = f is the number of scores. Find the mean ( xr ) of the following data sets a x x= r = n = 36 9 b x xr = = n = = 4 = 43. The Median is the middle score of an ordered set. To find it, arrange the data in ascending order and pick the middle score. If there is an even number of scores, then the median is the average of the two scores in the middle. If n is odd, then the median is the score in position n + 1. If n is even, then the median is the average of the scores in position n and n + 1. Find the median of the following data sets a b Arrange in ascending order Arrange in ascending order middle score middle two scores ` The median is 4. ` The median is 4+ 5 = 4.5. The Mode is the score with the highest frequency. The Mode is the score that appears the most. A set could have no modes (equal frequency for all scores) or have more than one mode (more than one score with the highest frequency). Find the mode of the following data sets a b 3 and 4 have the highest frequency () 7 has the highest frequency (3) ` 3 and 4 are modes of this data set. ` 7 is the mode of this data set. These 3 statistics have a fancy name. They are called measures of central tendency % Data
2 Finding the Mean, Median and Mode from a Frequency Table Sometimes a data set could be very large and so we would use a frequency table to find the measures of central tendency. Look at the table below: Score (x) Frequency (f) fx Cumulative Frequency (cf) 6 # 6 = # 4 = # 7 = # 9 = # 4 = 3 30 n = f = 30 fx = 154 The third column is a new column. It contains the product of the score and its frequency. The mean of data in a frequency table is xr = fx f To find the median use the cumulative frequency to determine the score in the middle position. To find the mode, just identify the score with the highest frequency. Use the data set above to find the mean, median and mode of the data set The mean for the above data set is xr In the table above: fx = = 154 f 30 = 5.1 f = 30. So there are 30 scores in the data set. This is an even number. ` the median will be the average of the scores in 15 th and 16 th position. Since the cumulative frequency of 5 is 17, it means that the score in 15 th and 16 th position is 5. ` the median is 5+ 5 = 5 In the table above, 7 has the highest frequency (9). So the mode of the data is 7. The range of a set of data is the difference between the highest score and the lowest score. So, for the above table: Range = HighestScore -Lowest Score = 8- = 6 Mathletics 100% 100% Data 3P Learning SERIES TOPIC 7
3 1. Answer these questions in your own words. a What is the mean of a data set? b What is the median of a data set? c What is the mode of a data set? d What is the range of a data set? e What do f and fx mean? 8 100% Data
4 . A teacher records the percentages that a group of students achieved in a test a Find xr, the mean of the data set. b Arrange the data set in ascending order. c Find the median of the data set. d Find the mode(s) of the data set. Mathletics 100% 100% Data 3P Learning SERIES TOPIC 9
5 3. Complete the following frequency distribution table. Score (x) Frequency (f) fx Cumulative Frequency (cf) a What is the range of this data set? 1 8 n = f = fx = b Find xr. c How many scores are in the data set? d In which position is the median score and what is the median? e What is the mode of the data set? % Data
6 4. Look at this data set: a Find xr. b If an 11 th score of 6 was added to the data set, is the new mean greater than, equal to or less than the original mean? Why? c If an 11 th score of 5 was added to the data set, is the new mean greater than, equal to or less than the original mean? Why? d If an 11 th score of 4 was added to the data set, is the new mean greater than, equal to or less than the original mean? Why? Mathletics 100% 100% Data 3P Learning SERIES TOPIC 11
7 5. This data set shows how many points a team scored each week over a 4 week period: How many points do they need to score in the 5 th week to ensure that the mean score over 5 weeks is 00 points? 6. How many scores are in a data set if: a The median is the 17 th score in an ordered data set? b The median is the average of the 0 th and 1 st score in the data set? 1 100% Data
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