Chapter 5 Statistical Reasoning 5.1 Exploring Data
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1 Chapter 5 Statistical Reasoning 5.1 Exploring Data Nov 20 8:04 AM Statistics the branch of applied mathematics concerned with the collection, analysis and interpretation of numerical data. When data is collected, it must be presented in such a way that comparisons can be made. Data can be collected from a population or from a sample of the population Once the data has been collected, it can be analyzed in a variety of ways. Measures of Central Tendency Values that describe how data is grouped or clustered around the center. Nov 20 8:05 AM 1
2 Three Measures of Central Tendency: MEAN, MEDIAN, MODE 1. Mean Commonly called the average There are two types "population mean" or "sample mean" Both are calculated the same way! Steps: 1. Add all the data values together 2. Divide by the number of data values you have Example: Calculate the mean of 8, 20, 33, 55, 67 Nov 15 6:42 PM 2. Median The data value that lies in the middle after putting all the data in order BE CAREFUL there are two cases to consider! Steps with an odd number of data values 1. Write numbers in order 2. Select the middle number Example: What is the median of 8, 20, 33, 55, 67? Steps with an even number of data values 1. Write the numbers in order 2. Select the two middle numbers 3. Determine the mean of these two numbers Example: What is the median of 9, 10, 14,19, 20, 24? HINT: To determine the middle position Middle position = ( # data values + 1 ) 2 Example: What would the position of the middle value be if you had 503 pieces of data? Nov 15 6:59 PM 2
3 3. Mode The number that occurs most often It is possible to have one, two or no mode Steps 1. Write the numbers in order 2. Count which number is most popular Example: What is the mode of the following list of numbers? 1, 2, 2, 5, 6, 12, 12, 12, 18, 19 Nov 15 7:12 PM Nov 15 7:20 PM 3
4 When comparing data it is also important to look at how the data is dispersed or spread out. Dispersion A measure that varies by the spread among the data in a set. If the set of data is identical, the dispersion value is 0 As the data is spread out, the value of the dispersion increases. Example: i) 5.4, 5.4, 5.4, 5.4 : has a dispersion = 0 (identical) ii) 1, 15, 1000: dispersion is high (the data is spread out.) Range A way to measure dispersion. Range = the highest data value the lowest data value Example: Determine the range of 5, 5, 10, 10, 10, 15, 15, 15, 20, 25 Outlier A value in the data set that is very different from other values in the set. Nov 20 8:35 AM Ex 1: Tim and Luke are both enrolled in Mathematics 2201 and scored the following marks on the last five unit tests. a) Determine the measures of central tendency (mean, median, and mode) for the test marks for each student. b) Calculate the range values for the test marks for each student. What can be learned from these values? Nov 20 8:38 AM 4
5 Ex. 2: If the mean of the data 12, 6, 12, 10, 7, x, 11, 9 is nine, find the value of x. Nov 15 7:39 PM Ex. 3: The data below represents the number of home runs hit by Ricky during his summer rec league a) Calculate, to one decimal place, the mean, median and mode for the number of homeruns per game. b) Annabelle drew a frquency table to represent the data. Show how Annabelle can use the data in the frequency table to calculate the mean, median, and mode for the number of homeruns per game. Nov 20 12:06 PM 5
6 Ex 4: Calculate the range of each group below. Explain why the range, by itself, can be a misleading measure of dispersion. Group A: 8, 13, 13, 14, 14, 14, 15, 15, 20 Group B: 7, 7, 8, 9, 11, 13, 15, 15, 17, 18 Nov 20 8:42 AM Nov 15 6:11 PM 6
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