3.2 Measures of Central Tendency Lesson MDM4U Jensen

Transcription

1 3.2 Measures of Central Tendency Lesson MDM4U Jensen - In this section, you will learn how to describe a set of numeric data using a single value - The value you calculate will describe the of the set of data - The 3 measures of central tendency are: The following examples outline situations in which each measure of central tendency is most useful The Mean The mean: The formula for the mean of a set of values is: The Greek symbol Σ (sigma) indicates that all the values of x in a set of data are added together. The sum is divided by the number of values in the set, n. Example 1: Find the mean of the following set of values:

2 What does the mean.mean? The mean can be understood as the centre of gravity if we examine the following situation involving weights and balances: - The location of the triangular fulcrum is the centre of gravity which is the same as the arithmetic mean. - Notice that the sum of the distances from the fulcrum is the same on the left and right side. - In stats, this distance is called the from the mean. If you consider distances to the left of the fulcrum as negative, then the mean is the value that makes the sum of the deviations from the mean equal to zero. Weighted Mean In general, the weighted mean can be calculated as: Example 2: What if those blocks on the ruler were different weights? 2 cm block is 10 grams 18 cm block is 15 grams 25 cm block is 12 grams Where should the fulcrum be now?

3 Example 3: MDM4U Course Breakdown: Weight Mark Assignments 15% 92 Tests 40% 86 ISU 15% 85 Exam 30% 88 If these were your grades, what would be your final mark? Mean of Grouped Data Supposed your data have already been organized into a frequency table with a class interval not equal to 1. You no longer have actual data values, so you must then use the of each class to estimate a mean weighted by the frequency. Finding the average (mean) of grouped data is the same as finding a weighted average; except that you have to use the interval midpoint as the data value. Example 4: A sample of car owners was asked how old they were when they got their first car. The results were then reported in a frequency distribution. Calculate the mean. The formula you will use is: Where m is the interval midpoint and f is the frequency of the interval.

4 Start by completing the table: Now use the formula to calculate the mean: The Median The median value is the data point in an set, dividing the set into two sets of equal size. If the set has an even number of data points, then the median is halfway between the two middle- most values. Example 5: Monthly rents downtown and in the suburbs are collected from the classified section of a newspaper. Calculate the median rent in each district Downtown: 850, 750, 1225, 1000, 800, 1100, 3200 Suburbs: 750, 550, 900, 585, 220, 625, 500, 800 Start by ordering the sets of data Downtown: Suburbs:

5 Downtown There are elements in the set, so the median is the element. The median rent downtown is /month. Suburbs There are elements in the set, so the median is halfway between the th and th elements. Halfway between these elements is: Therefore, the median rent in the suburbs is /month. The Mode The mode is simply the value or range of values in a data set. It is easy to determine the mode from a histogram as it is the highest column. Example 6 The mode of the histogram shown below is: Note: If no measurement is repeated, the data set has no mode. If it has two measurements that occur most often, it is called bimodal.

6 How does the shape of a distribution affect the measures of central tendency? The distribution is shaped and symmetric. The mean median and mode are all. The distribution is skewed. It is asymmetric. Notice that the mode < median < mean. This happens because the outliers affect the mean more than they do the median and mode. c) What if the data was skewed left? Label the mean, median and mode approximately.

7 Skewed left: The mean is the median Skewed right: The mean is the median Symmetrical The mean, median, and mode are. What Measure Should You Use? With Outliers: With a Strong Skew: Mound Shaped:

8 Qualitative Data: Before you start your homework... Your marks... weight mark tests 45% 86 assignments 15% 92 ISU 10% 85 exam 30%? a) If these were your marks, what would your term mark be going into the exam? b) What mark must you achieve on the final exam to earn a final grade of 85%? Homework: Pg. 158 #1-5, 7, 9, 10, 11, 12, 13 #17 is extra practice for mean of grouped data