PS2: LT2.4 6E.1-4 MEASURE OF CENTER MEASURES OF CENTER

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1 PS2: LT2.4 6E.1-4 MEASURE OF CENTER That s a mouthful MEASURES OF CENTER There are 3 measures of center that you are familiar with. We are going to use notation that may be unfamiliar, so pay attention. 1. Mode: The most frequent or highest frequency Bimodal: 2 modes for a data set. For grouped numerical data, we talk about a modal class, which is the class that occurs most frequently. 2. Median: Middle most term when the data is organized from least to greatest. 3. Mean: The average score by taking the sum of all scores and dividing them by the total frequency of the values. GaEyL_eXATof04d6Iw&safe=active 1

2 SIGMA NOTATION Summation notation or sigma notation is used to show the sum of values. It is the sum of the finite set. n=total data points (ending) i=1 (starting point) x i (terms in the summation) = the sum of all x i values Lets break it down, using the set v = {1,3,4,9} n i=1 v = = 17 WHY BRING THIS UP? We use this notation to find the mean. Suppose x is a numerical variable and there are n data values in the sample. We let x i be the i th data value from the sample of values x 1, x 2, x 3, x n 1, x n. The mean of the sample is denotes by x bar or x. n x = x 1 + x 2 + x 3 + x n n i=1 = n x i Note: We denote μ mu as the population. 2

3 MEDIAN The median cuts the data in halves. Half of the data are less than or equal to the median, and half are greater than or equal to it. DON T FORGET TO ORDER THE DATA FIRST BEFORE FINDING THE MEDIAN!!! For an odd number of data, the median is an actual data point. For an even number of data, the median is the average of the two middle values, and may not be in the original data set. If there are n data values, the median is the n+ 1 th data value. 2 EXAMPLE What term would be the middle of a data set with n = 39 n+1 = 39+1 = 40 = 20. So the median is the th ordered data value. What term would the middle of data set with n = 1344 be? 3

4 FIND THE MEASURE OF CENTER! A teenage delinquent recorded the time (in minutes per day) he spent playing computer games rather than doing his homework over a 2 week holiday period: 121, 65, 45, 130, 150, 83, 148, 137, 20, 173, 56, 49, 104, 97. Lets use the calculators stat edit enter the values in L 1 stat over to calc 1 var stats calculate (some calculators ask for the list which is 2nd 1) CHOOSING THE RIGHT MEASURE OF CENTRE Mean: Commonly used and easy to understand Takes all values into account Affected by extremes (outliers). (do not use mean when data has an outlier) Mode: Gives the most usual value Only takes common values into account Not affected by extremes (outliers) Median: Gives the halfway point of the data Only takes middles values into account Not affected by extremes (outliers) 4

5 MEASURES OF THE CENTRE FROM OTHER SOURCES Frequency tables can make the measures of center more difficult, however, it is better then the alternative. Mode: The most frequent. Median: middle. How would we find the middle term? Mean: This can be tricky, but there is a way to do this using the table. MEASURES OF THE CENTRE FROM OTHER SOURCES Mean: We can sum up the data value multiplied by the frequency, then divided by the sum of the frequencies. So, can someone interpret: x = fx f 5

6 MEASURES OF THE CENTRE FROM OTHER SOURCES Try it find it on your own. x = fx = 278 f THE MEDIAN What is the middle term? 20th and 21st data values. We can create another column for the cumulative frequency, which is the frequencies added together starting with the lowest data value, to the highest. Cumulative Frequency Therefore the middle term must be

7 EXAMPLE The table shows the number of aces served by tennis players in their first sets of a tournament. Determine the: A. mean B. median C. mode for this data. A) 3.25 aces B) the 28 th data value is 3 aces. C) 3 aces. THE OTHER APPROACH Using a calculator, lets do another example: Find the mean, median and mode. IA Scores Frequency

8 GROUPED DATA Estimate the mean of the following ages of bus drivers data, to the nearest year: We need to organize this in a vertical frequency table, and use the midpoint of the grouped data to use as x. mean 37.7 HOMEWORK 6E.1 #1,4,12,13,16 6E.2 #2,3 6E.3 #2-4 6E.4 #1,2,5,6 8