Measures of Central Tendency . Block 46 Module 1. Welcome & Overview. Examine Next Generation Sunshine State Benchmarks related to Measures of Central Tendency Administer Pretest Introduce the module objectives Developmental Activities: Measures of Central Tendency – Mean, Mode, Median
Vocabulary development, problem solving, technology, error analysis/misconceptions
?Measures of Central Tendency?
Describe and complete an activity that may be used to build student vocabulary power.
Terms: central tendency, mean, median, mode, variability, range, set of data, frequency table, numerical data, categorical data, outliers, line plot, circle graph, continuous data,
categorical data, common, middle,
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“Manipulating the physical model not only helps [children] understand the formula but also promotes retention.”
“Simply being able to state the algorithm for finding these statistics is not enough. To support the development of data sense, each of these should be developed meaningfully through concrete activities before introducing computation.
(Reys, et. al. 2009; p. 395)
Find the median of 2, 3, 4, 2, 6.
Participants will use a strip of grid paper that has exactly as many boxes as data values. Have them place each ordered data value into a box. Fold the strip in half. The median is the fold.
i) the mean,
ii) the median,
iii) the mode?
Fold this in half (divide by 2).
Purpose: Reinforce concepts through songs.
Click in the link
Cheers: Mean, Median, and Mode
Mean (Say in really mean voice and face throughout!)Add all the numbers (Have hand go from waist to neck in increments.)And Divide! (Have same hand slice across the neck!)
Median …MiddleOrder numbers least to greatest (motion hand left to right)Find the middle. (move both hands to middle and clap)
When I say mode you say MostMode….MostMode….Most!Mode is the number that appears most oftenMode…Most oftenMode…Most often!
add all of the values, then divide by the number of values.
8 + 5 + 9 + 6 = 28
Mean score on the math test is 7
8, 5, 9, and 6 or 5, 6, 8, 9
5, 6, 8, 9
(6 + 8) ÷ 2 = 7
Median score on the math test is 7
8, 5, 9, and 6 or 5, 6, 8, 9
list all of the values, in order of size. Select middle value(s)
M E D I A N
Finding Mean, Median and Mode for data sets
Give the remaining seated students a calculator.Follow directions on activity sheet.
Discuss how the outlier affects the measures of central tendency.
Using different Strategies!
Andy’s results on three tests are: 68, 78, and 88.
1. Find the mean and median score.
2. Explain why the mode is of little value.
3. What score would be needed on the next test to get an average of 81.
4. Describe two different ways you could determine this score.
MA.8.S.3.2: Determine and describe how changes in data values impact measures of central tendency.
Mrs. Jensen’s 7th grade class was surveyed about how much allowance each student receives each week. The results are shown in the table.
Use this information to make a spreadsheet for the data, and find the mean, median, and mode.
Analyze the results.
Research into Practice
“Many middle graders are able to calculate averages but their understanding of the concept of average is shallow.” (Reys, et al., 2009).
Participants will identify misconceptions students might have about statistics.
Purpose: to review measures of central tendency, and provide an opportunity for participants to use research findings to inform their instructional practices.
“Akira read from a book on Monday, Tuesday and Wednesday. He read an average of 10 pages per day. Circle whether each of the following is possible or not possible.”
“Many middle graders are able to calculate averages but of understanding of the concept of average is shallow.” (Reys, et al., 2009, p. 396).
Work with a partner to develop an activity to help students better understand the concept of average.
Present your activity to the class with a rationale for selecting this activity.
Problem: When finding the median of an even-numbered set of data, some students use the mean the data instead of the mean of the two middle numbers.
Describe an activity that may be used to help students overcome this problem.
Make a list of the different types of graphs that you know. Select one of the graphs, and write about a situation in which you will use that graph to display data.
Participants compare two Bar graphs with the same data, and discuss why one may be misleading.
Which graph would you use to compare the number of red folders sold by two stores in one week?”
Given the data set: 7, 51, 25, 47, 42, 55, 50, 26, 44, 55, 26, 33, 39, participants will:
Copy the numbers unto individual index card.
Sort the cards into piles based on place value. (Stem value)
Note what they have in common
Cut one of the stem, place it on a sheet of paper or the table. Cut the remaining leaves with this stem. Add the leaves to the leaf section of the plot.
Repeat for each of the piles.
Discuss the plot by examining the lowest and highest scores, the lengths of each leaf, gaps, tapers, and the median and modal values.
An example of a stem-and-leaf plot for the data set (34, 30, 38, 42, 67, 68, 68, 56, 54, 34, 82, and 85) is as follows: Legend: 3|234
Measures of Variation
Examine web and ask:
Mean, the median and the mode are measures of the central tendency of a set of data.
However, such measurements cannot tell us the spread or variation of the data.
Dispersion is the statistical name for the spread or variability of data.
Lower QuartileUpper Quartile
2 3 5 7 10 11 15 16
find the lower median. This is the middle of the lower six numbers. The exact centre is half-way between 8 and 9 ... which would be 8.5Now find the upper median. This is the middle of the upper six numbers. The exact centre is half-way between 14 and 14 ... which must be 14
quartile. This is the
middle of the lower
six numbers. The exact
center is half-way
between 8 and 9 ... (8.5)
This is the middle of
the upper six numbers.
The exact centre is
half-way between 14 and
14 ... which must be 14
If the greatest or the least value (assuming both are unique) in a data set is removed, then
If a zero value is inserted in a positive data set, then
A box and whisker graph is used to display a set of data so that you can easily see where most of the numbers are.
What are the median, lower and upper quartiles?
Median Upper Q
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Where do the numbers cluster together and where do they spread out? Explain.
Where are the extremes located in relation to the clustered data?
The difference between the two end-points of the line (represented by the highest and lowest marks) is the range.
The length of the box is the inter-quartile range.
Double Box and Whisker Plots
Alfonso's bowling scores are 125, 142, 165, 138, 176, 102, 156, 130, and 142. Make a box-and-whiskers plot of the data. The box and whiskers plot below represents the bowling scores of Anna. Compare the bowling scores of Alfonso and Anna. Who is a better bowler?
Investigation: Drops on a Penny: Graphing Data
It is a scattered plotting of points that may or may not seem to follow some sort of trend or pattern.
A fitted line (or line of best fit) of course!
The fitted line is one that is closest to every point in the plot.
If two points are a ways a part, the line must compromise and go between them.
In groups of three, examine the following sample scatter plots. Once you have ’thoroughly’ examine them, rank them in order from strongest to weakest relationship between the variables. Finally, briefly explain your selections in the area provided below.
Be ready to present your findings!
3 Types of Relationships
Draw the graph to show the correlation between shoe size and height.
This is one of the two things scatter plots can be used to find the answer to:
Reflections & Suggestions