Unit 8: Statistics Lesson #5: One-Variable Statistical Analysis June 6th, 2011

There are two basic types of data that can be analyzed: Categorical Data – This is data that can be divided into groups (i.e. Smarties can be divided into groups based on colour). Numerical Data- This is data that consists of numbers that correspond to some property of an item (i.e. the number of Canadians that use the internet daily for at least 2 hours).

Statistical data can be represented in many graphical forms that we have already looked at this year. For example, Pie/circle graphs, histograms, pictographs and bar graphs are all commonly used to represent data. After the data is graphed, we use those graphs to look for characteristics of the data. Statistics are just numbers used to describe sets of data

Two types of statistics that you need to know are: Measures of central tendency and Measures of dispersion and we have talked about some of these a little already.

Measures of central tendency: The centre of the data is the value to which most of the data are reasonably close. There are three measures of central tendency that are commonly used: the mean (average), the median, and the mode. The mean (average) is represented by the symbol x. To find x use the formula:

Example 1: The numbers of service calls a heating company made during the first eleven days of November are listed below. Find the mean for this set of data: 6, 28, 28, 11, 30, 21, 17, 28, 28, 20 SOLUTION – Order the data: 6, 11, 17, 20, 21, 28, 28, 28, 28, 30

Find the mean:

The median represents the middle number in an ordered set of data (arranged in ascending order). If the set of data is odd numbered, then the median number will be part of the data set. However, if the set of data is even numbered, then the median is the average of the middle two elements. Example 2: The numbers of service calls a heating company made during the first eleven days of November are listed below. Find the median of this set of data. 6, 28, 28, 11, 30, 21, 17, 28, 28, 20

SOLUTION: Order the data 6, 11, 17, 20, 21, 28, 28, 28, 28, 30 There are the values, so the middle value (the median) is between value 5 and 6. To find: MEDIAN = 21+28/2 = 24.5

The mode represents the most frequently occurring value The mode represents the most frequently occurring value. If there is no repeating value in the data, there is no mode. If two values are repeated the same number of times, then there are two modes. Example 3: The numbers of service calls a heating company made during the first eleven days of November are listed below. Find the mode of this set of data. 6, 28, 28, 11, 30, 21, 17, 28, 28, 20 SOLUTION: Order the data and find the most common value: 6, 11, 17, 20, 21, 28, 28, 28, 28, 30 In this case the mode is 28

Measures of Dispersion: Once you have determined the centre value of a set of data, it is important to know how the data is spread out or dispersed. The most commonly used measures of dispersion are the range, percentiles or quartiles, and standard deviation. We've already learned how to find the range and quartiles of an ordered set of data.

Standard Deviation: This is the most commonly used measured of dispersion. It is denoted by the symbol and it tells us how much variation there is or how “dispersed” our data is from the mean. The standard deviation is found by taking the square root of the sum of the average squared difference of each value from the mean. Use the formula: Where x = value = mean n = number of values

Example 4: Find the standard deviation of the set A = {1, 3, 7, 12, 18, 19}. = Mean = 60/6 = 10

This means that one standard deviation is 6. 93 This means that one standard deviation is 6.93. Most of the data falls within 6.93 units of the mean.

Using a Graphing Calculator 1. Press STAT, Enter to access the list editor. 2. Enter the values in Set A into L1. 3. Press STAT and use the arrow key to select CALC menu. 4. Press 1 for 1-Var Stats. 5. Press ENTER.