Joseph A. Montagna
The statistical calculations will we employ in this unit will be limited to measures of central tendency. A
measure of central tendency
is an index of central location employed in the description of frequency distributions. The only measures we will learn about in this unit are:. mean, median and mode. Our study of these measures will be basic, leaving out the more complex and sophisticated descriptions involving dispersion of data. We will simply involve ourselves in finding the central values for the data collected.
1.1 THE ARITHMETIC MEAN
Time allotted: one class period plus homework assignment
The
mean
is a measure of central tendency that acts like the fulcrum on a seesaw. The mean balances the scores on either side of it. The way to calculate the mean is to find the sum of the scores or values, then divide the sum by the number of scores. The algebraic form is:
(figure available in print form)
X = the mean (“X bar”)
N = the number of scores
· = the mathematical verb which tells us to sum the scores
Give students a set of scores to find the mean of:.
14, 24, 29, 19, 12, 21, 17
The sum is 136. Dividing 136 by 7 gives us 19.4285
We can leave the number as is, or we can round it to any of several places.
Give students enough practice in class, and assign a suitable homework.
1.11 THE EFFECT OF EXTREME SCORES ON THE MEAN
Time allotted: one class period plus homework
Above we described the mean as a fulcrum of a seesaw. Ask students about how to balance a seesaw that has one person who is heavier than the person on the opposite end of the seesaw. The answer is that the heavier person has to sit closer to the fulcrum, or the lighter person must move away from it. When working with numbers, how does an extreme measure or score effect the mean?
Take the set of scores that were used as an example in 1.1, and add an extreme score to it Below is that set with 100 added to it.
14, 24, 29, 19, 12, 21, 17, 100
Now have students find the mean of this set. 236 = 29.5
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8
This second mean is approximately 50% larger than the first by adding the extreme value to the set. The mean is very sensitive to extreme scores. Try this again with the first set adding a low extreme score to it.
Give practice in class in calculating the means of sets which have extreme scores in them. One suggested way to do this is to use some of the previous lesson’s work, adding extreme scores to them.
1.2 THE MEDIAN
Time allotted: one class period plus homework
The
median
is that score or value which has an equal number of scores on either side of it. The median is the exact middle of a set of values. The 5Oth percentile is the median. (If an odd number of scores exists in a set, then the middle number is the median. An even number of scores requires finding the mean of the scores on either side of the middle.)
Ordering the scores from 1.1 gives us: 12,14,17,19,21,24,29
The median is 19. Three scores lie on either side of 19.
1.21 THE EFFECT OF EXTREME SCORES ON THE MEDIAN
Time allotted: incorporate into the previous lesson
Does adding an extreme score have any effect on the median? Let us use the scores from 1.11:
12,14,17,19,21,24,29,100
Because this set of numbers has an even number of elements (8), we have to find the mean of the two numbers on either side of the middle, 19 and 21. The median now is 20. In this case the median is effected only slightly.
Give students an opportunity to check this out in a few other cases.
1.3 THE MODE
Time allotted: one class period
The
mode
is very easy to identify in a given set of values. It is the value which appears with the greatest frequency in a set of values. In the case of the sets previously mentioned there is no mode. This happens occasionally.
Give students practice in finding the mode of sets of values.
2.0 CLASS PROJECT
Decide upon a topic for a class project which will yield a set of values that students can use to practice finding mean, median and mode.
Suggested topic: height of students in class
Time allotted: one class period
Measure each student’s height in centimeters.
Put the individual heights in order, from tallest to shortest.
2.1 Find the mean height of the class (one class period)
2.2 Find the median height of the class
2.3 Find the mode (if one exists)
2.4 Discussion of findings
Time allotted: one class period
Refer to the findings in the lessons involving the class project mentioned above. Draw students into a discussion of how these three measures of central tendency relate to one another. How do they differ?
Have students order themselves by height in front of the class.
Which student is in the exact middle? Is this person the median or the mean?
Refer to the mean value obtained. Which person(s) are that height? Which persons represent the extremes in this class?
2.5 GRAPHING
Time allotted: one class period
Have students to construct histograms of the data collected in this project. The teacher should construct a larger graph of the sane data to be used for discussion purposes and instructions on how to construct a histogram. (Students need to decide how to label their intervals on the graph. They should discover that the data involved will determine the size of each interval.)
3.0 CLASS PROJECT: DATA COLLECTION & DISPLAY TECHNIQUE
Time allotted: Three or four class periods plus homework
Discuss some simple ideas that students can begin to pursue with classmates. Students often begin with simple survey questions concerning “favorite”, e.g. What is your favorite type of sneakers? It is desired that they start with simple topics. The objective is for the students to learn data collection and display of that data. Certain skills are presumed to be held by the students. If the student does not possess these prerequisites, they should be taught first. The following activities require the following skills: knowledge of fractions, division involving decimals, finding percentages, rounding, general knowledge in the use of protractors and compasses.
Build a list on the blackboard that includes students’ suggested topics. Each student should have one of his/her own. Ex: “What is your favorite automobile?”
Have students construct a data collection sheet similar to the one shown. They should begin data collection as soon as they have finished constructing the data collection sheet. They will certainly have to finish outside of class. Collection is facilitated for this lesson by students verbally asking the question.
(figure available in print form)
(Tally marks are used in the column next to responses. Note that students put a tally mark after first recording one particular response.)
Upon completion of collecting the raw data, discuss some display methods one may use (histogram. pictograph, “pie” graph).
For the purposes of this lesson we will employ the pie graph method. This method is a personal favorite of mine because it requires the use of a number of math subskills.
Before actually constructing the pie graph students will have to properly fill in the next three columns on the data collection sheet.
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-Fraction column The count becomes the numerator of the fraction. The denominator is the total number who were surveyed.
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-Decimal column Dividing the denominator into the numerator yields a decimal fraction. If This decimal is taken to the thousandths place.
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-Percent column Students can round the decimal column to the nearest hundredth. Dropping the decimal point and adding the percent sign produces the percent figures.
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-Degrees column Multiplying the decimal column figures by 360 (number of degrees in a circle) yields the size of each “slice” of the pie graph. Students should round to the nearest whole degree. Note that the total number of degrees may add up to more than 360 due to rounding.
Construct the pie graph using the information they compiled on their data collection sheets.
Present results of individual surveys to entire class.
It is recognized that this lesson will require more time than the normal class period. While some of this may be completed outside of class, students will need close supervision by the teacher. Two class periods with homework assigned may be enough time.
(figure available in print form)
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Try to engage students in a discussion of the relationships of the numbers they see in this chart. Can they discover that a fraction that is twice another fraction yields numbers that are twice as large in the remaining columns, i.e. 1/26 as compared to 2/26. Can they understand the relationship between fractions, decimals and percentages? There is much to be gleaned from this lesson if the time is put into it.
4.0 CLASS PROJECT: Leisure Time Activities
Time allotted: Two class periods plus homework
We are now going to conduct a study of what your classmates like to do during their leisure time. On the list given to you are to “prioritized” the activities mentioned, giving a “1” to what you like to do best, “2” to your second choice, etc.
Collect data in class. Have students copy down the priority numbers that you read to them off of the students’ questionnaires. Their homework assignment is to find the mean value for each activity.
4.1 Discussion of homework findings
Time allotted: one class period
First see that everyone agrees on the means for each activity. No doubt there will be discrepancies. Clear these up as you go along. The priority values will likely be in decimal form, making it an added challenge for students to put the means in order. The lowest value is the activity that the class has as a priority leisure activity. Have students to compare their own priority values to those of the class.
(an alternative method is to use small groups calculate the means)
5.0 CLASS PROJECT: Survey on clique formation
Time allotted: Two class periods
In the early sixties James Coleman asked a number of high school students the following question, “What does it take to get into the leading crowd in your school?” It would be interesting to find out how your students would rank the following areas:
WHAT DOES IT TAKE TO GET INTO THE LEADING CROWD
IN YOUR SCHOOL?
__ personality
__ be an athlete
__ good grades
__ good looks
__ have money
__ good clothes
__ come from right neighborhood
5.1 Collect data from 5.0. Calculate mean of each. Order list
Time allotted: one class period
See lessons 4.0 & 4.1
