3.2 Measures of Central Tendency - 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 centre of the set of data - The 3 measures of central tendency are: 1. Mean 2. Median 3. Mode - the following examples outline situations in which each measure of central tendency is most useful
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3.2 Measures of Central Tendency
- 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 centre of the set of data- The 3 measures of central tendency are:
1. Mean
2. Median
3. Mode
- the following examples outline situations in which each measure of central tendency is most useful
The Mean
The mean: a measure of central tendency found by dividing the sum of all the data by the number of pieces of data.
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:
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:
2cm 18cm 25cm
15cm
- 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 deviation 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.
2cm 18cm 25cm
15cm
Weighted Mean
In general, the weighted mean can be calculated as:
Where x represents each data value and w represents its weight, or frequency.
Example 2
What if those blocks on the ruler were different weights?
2 cm block is 10 grams18 cm block is 15 grams25 cm block is 12 grams
Where should the fulcrum be now?
2cm 18cm 25cm15.9cm
Example 3
MDM4U Course Breakdown:
Weight MarkAssignments 15% 92Tests 40% 86ISU 15% 85Exam 30% 88
If these were your grades, what would be your 3inal 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 midpoint 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.
Start by completing the table:
Now use the formula to calculate the mean:
The median value is the middle data point in an ordered 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.
The Median
Example 5Monthly 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, 3200Suburbs: 750, 550, 900, 585, 220, 625, 500, 800
Start by ordering the sets of data…
Downtown:
Suburbs:
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 ModeThe mode is simply the most frequent 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.
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.
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.
ModeMean
Median
c) What if the data was skewed left? Label the mean, median and mode approximately.
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?
Median is most effective
Mean is most impacted by outliers
Median is most effective
with outliers:
with strong skew:
mound shaped (symmetric):
all three will be close, so any are appropriate
Qualitative Data:
Mode is most appropriate if the data is not numeric.
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%?
Before you start your homework...
Pg. 158 #1-5, 7, 9, 10, 11, 12, 13
#17 is extra practice for mean of grouped data
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