Measures of Central Tendency

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Amberlynn Patterson
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1 Measures of Central Tendency Here we will look at some statistics that summarize certain features of the data. These statistics are mode, median and mean. We have already encountered mode and median. Mean is a measure which we commonly called "average." Mean is the summation of all data divided by the total frequency or observations. For discrete data x, the MEAN = x where n is the n total number of observations. If we are dealing with a set of data x then x is used as the symbol for the mean of x. In fact, x represents the sample mean. Population mean is represented by " μ ". Recall that a sample is PART of a population. If our sample size is very large, then we have a reasonably good estimate of the population mean using the sample mean. Examples: Find the mean, median, and mode of the following data representations: a. Mary carries out a survey about the number of horses in 7 farms. Here is her raw data b. A group of students are asked how many books they had read in the previous week. The results are represented in a frequency table below. Number of books, x Number of students, f

2 c. The bar chart below shows the results of a group of students who were asked how many movies they had watched in the previous week. Solutions: a. We first need to rearrange the data in increasing order The mode is 6 because it has the highest observed frequency. The median is 6 and is in the "middle" of the ordered distribution The mean = = 6.29 (3 s.f). 7 b. Number of books, x Number of students, f (x)(f) The mode is 2 because it has the highest frequency.

3 There are 45 students. Half of 45 is 22.5, so the value (observation) of the 23rd student is the median. To arrive at the 23rd student we would have to add up the frequency. We obtain = 16 and = 36, that is up to the 16th student 1 book is read. The 17th to 36th student each read 2 books. Therefore, the 23rd student must have read 2 books. So the median is 2. xf sum of products =. f sum of frequencies The sums are in italic bold above, = 1.84 (3 s.f). c. The mode is the tallest bar. 1 is the answer. There are ( ) = 32 students. Half of 32 is 16. The median is the value between the 16th and 17th observation. Again, we need to add up the frequencies. 8 people watched no movies. The next 10 people up to 18th observation watched 1 movie. Thus, both the 16th and 17th observations are just 1 movie. The median is 1. xf sum of products =. f sum of frequencies [(0x8) + (1x10) + (2x6) + (3x8)] = 1.43 (3 s.f). 32

4 Use of GDC. [STAT] Select 1:EDIT [ENTER] {enter your data x into the column under L1 and frequencies into another column under L2} Once finished, then press [2nd][MODE] for quit. Press [STAT] again select CALC 1:1-Var Stats [Enter] You will then see this in your screen: 1-Var Stats {blinking cursor} Enter [2nd][1] [,] [2nd][2] for "L1,L2" and press [ENTER]. Recall that the median is Q2. Note that the order here is important. You should always enter the list that contains x first, followed by a comma (which is the command for multiplication in this little routine) and then your frequency list. Summary: a. If the total number of observations is n, then the value of the (n/2)+(1/2)th observation is the median. b. If n is 15 then the value of the (15/2)+(1/2)th = 8th observation is the median. c. If n is 6 then the value of (6/2)+(1/2) = 3.5 then the median is the value that lies between 3rd and 4th observations.

5 Mean Advantages 1. Easy to calculate. 2. It makes use of the complete data. 3. It can be used in further statistics like standard deviation. Disadvantages It can be misleading if the data contains abnormally high or low values. Median 1. It has an easy interpretation. 2. Unaffected by very high or low values. 1. It has little usage in other statistics. 2. In grouped data, this has to be estimated from a cumulative frequency curve. 3. It is not a good summary of the data if the size is small or the distribution pattern is odd. Mode 1. It has an easy interpretation. 2. Unaffected by very high or low values. 3. A useful statistic for good manufacturers and sellers. It tells them the hottest selling item(s). 1. It has little usage in other statistics. 2. In grouped data, the mode cannot be determined with accuracy (refer to above examples). 3. It is not a good summary of the data if the size is small or the distribution pattern is odd.

What are the mean, median, and mode for the data set below? Step 1

What are the mean, median, and mode for the data set below? Step 1 Unit 11 Review Analyzing Data Name Per The mean is the average of the values. The median is the middle value(s) when the values are listed in order. The mode is the most common value(s). What are the mean,