CURRIKI ALGEBRA UNIT 3
Descriptive Statistics
Lesson 3.4: Summarize, Represent, and Interpret Data on a Single Count or
Measurement Variable:The Shape, Center and Spread of
Normal Distribution and Probability
Unit 3: Descriptive Statistics
The first four lessons (3.1-3.4) provide the instruction and practice that supports the culminating activity in the final unit project.
To the Teacher: This lesson is a bonus lesson and can be used if you choose.It provides extra practice on the shape, center and spread of normal distribution. This lesson might take two days to complete. The project for this unit takes two days to complete.
Lesson 3.4: Summarize, Represent, and Interpret Data on a Single Count or Measurement Variable: The Shape, Center and Spread of Normal Distribution and Probability
In this lesson, students will learn more about differences in the shape, center and spread of normal distribution and probability. They choose a summary statistic appropriate to the characteristics of the data distribution, such as the shape of the distribution or the existence of extreme data points.
Common Core State Standards by Cluster
Grade Level / Cluster / CCS Standard9 / Summarize, Represent, and Interpret Data on a Single count or measurement variable / S.ID.2, S.ID.3, S.ID.4
Lesson Preparation and Resources for Teachers:
YouTube Video of a probability machine demonstration
Sal Khan video, ck12.org Normal Distribution Problems: Qualitative sense of normal distributions
Mathematics, Chapter 44, Statistics, pages 21-23 (From Curriki, Siyavula)
Monty Hall Problem
Instructional Materials for Students:(print one copy for each student)
Olympics Distance and Time activity sheet for Scatter plot practice
Time: 50-minute session
Lesson Objectives:
Students will be able to:
- Review the mean, median, and mode of real-world data
- Observe randomness through watching a probability machine as related to a stock portfolio
- Review and apply what they know about probability
Lesson Content:
1. Background Building Activity for Students (10 minutes)
- Vocabulary Building
Ask students to use the Vocabulary Self-Collection Strategy (VSS) to learn math-specific words during the activities in this lesson. This activity is from Ruddell, M. R. (2005) Teaching content reading and writing. New Jersey, John Wiley & Sons, Inc.
Teachers initiate VSS by asking each student to note and nominate one math-specific word that they find during class, either in oral discussion or in writing, that they would like to know more about and that they think should appear on a class vocabulary list. The teacher also nominates a word. The students and teacher are required to tell the following about their word:
- Where they found the word. (Read the sentence if in a text or recall the context if it is from a discussion or activity.)
- What they think the word means in this context.
- Why they think the class should learn it (i.e., identify the word’s importance to the content topic).
At the end of the lesson, the teacher writes the words on the board and leads a discussion to define each, first from context as nominators tell what they think their word means, and then, if needed, from any references available in the room.Discussion should include contributions from other class members.
The list of words will vary, but might include: function, linear model, quadratic model, and expressions
b. Warm UpProblem
In real life, there are many things that illustrate normal distribution, including the grades and scores students get on tests. Use the example of students’ scores on a test.Most of the students score in the average range. A few students get 100% on the test and a few get less than 50% on the test.Draw this bell shaped curve on the board to illustrate. Another example is the height of 13-year-old students.Most of the students have heights that are average, but some are very short and some are very tall. This type of distribution can also be illustrated with a probability machine.
Problem 1
Show the whole class the YouTube Video that shows “Francis” the probability machine to see how random distribution usually falls in a bell shaped curve. (Click on the expansion symbol at the bottom right corner of the video to show with the full screen.)
YouTube Video of a probability machine demonstration
Let the students discuss their response to the video. Lead the discussion to the conclusion that a bell shape curve allows for the majority of the items to land in the middle of the curve and a few items to fall at either end of the curve, thus the “bell” shape.Ask students if they have any experiences with items in real life that fall on a bell shape curve.A few answers might include student grades on a test, the amount of people who like a certain kind of food, people’s weights at a give age, or the amount of home runs made on a Little League team. It is probable that we can get a bell shaped curve for many things in life.In today’s lesson we will see many applications of probability.
Problem 2
Normal Distribution Investigation
You are given a table of data below.
1. Calculate the mean, median, mode and standard deviation of the data.
2. What percentage of the data is within one standard deviation of the mean?
3. Draw a histogram of the data using intervals 60 ≤ x < 64, 64 ≤ x < 68, etc.
4. Join the midpoints of the bars to form a frequency polygon.
If large numbers of data are collected from a population, the graph will often have a bell shape. If the data was, say, examination results, a few learners usually get very high marks, a few very low marks and most get a mark in the middle range. We say a distribution is normal if
• the mean, median and mode are equal.
• it is symmetric around the mean.
• ±68% of the sample lies within one standard deviation of the mean, 95% within two standard deviations and 99% within three standard deviations of the mean.
What happens if the test was very easy or very difficult? Then the distribution may not be symmetrical. If extremely high or extremely low scores are added to a distribution, then the mean tends to shift towards these scores and the curve becomes skewed.
If the test was very difficult, the mean score is shifted to the left. In this case, we say the distribution is positively skewed, or skewed right.
If it was very easy, then many learners would get high scores, and the mean of the distribution would be shifted to the right. We say the distribution is negatively skewed, or skewed left.
Exercise: Normal Distribution
1. Given the pairs of normal curves below, sketch the graphs on the same set of axes and show any relation between them. An important point to remember is that the area beneath the curve corresponds to 100%.
a. Mean = 8, standard deviation = 4, Mean = 4, standard deviation = 8
b. Mean = 8, standard deviation = 4, Mean = 16, standard deviation =4
c. Mean = 8, standard deviation = 4, Mean = 8, standard deviation = 8
2. After a class test, the following scores were recorded:
- Draw the histogram of the results.
- Join the midpoints of each bar and draw a frequency polygon.
- What mark must one obtain in order to be in the top 2% of the class?
- Approximately 84% of the pupils passed the test. What was the pass mark?
- Is the distribution normal or skewed?
3. In a road safety study, the speed of 175 cars was monitored along a specific stretch of highway in order to find out whether there existed any link between high speed and the large number of accidents along the route. A frequency table of the results is drawn up below.
The mean speed was determined to be around 82 km.h−1while the median speed was worked out to be around 84,5 km.h−1.
- Draw a frequency polygon to visualize the data in the table above.
- Is this distribution symmetrical or skewed left or right? Why?
(From Chapter 44, Statistics, Grade 12:
2. Focus Question based on today’s lesson (25 minutes)
Focus Question:(Write it on the board.)
Probability: The Monty Hall Problem
Desired Learner Outcomes for Focus Question Activity:
Today we will be reviewing and applying what we know about probability and normal distribution through some fun activities.
a. Whole Class Activity:
Discuss probability and how that relates to real life.Ask students go give examples where probability comes into play.
A good example of probability is illustrated in the famous Monty Hall Problem.Show the class the YouTube video that discusses this famous problem.
Monty Hall Problem
Discuss Ben’s explanation of the probability problem using the three doors with a prize behind one.Ask students to work in groups of four to devise another problem that uses more than three doors.Ask the groups to describe their problem to the class and discuss the solution.
Students should be able to grasp the probability percentage and apply it through dividing 100% by the number of choices in their given problem, like Ben did with 33% in the move with three choices.
- Small Group Practice activity
Show theSal Khan video, ck12.org Normal Distribution Problems: Qualitative sense of normal distributions.In this video, Sal Khan describes normal distribution and presents a problem where you must choose items that would illustrate a normal distribution: The hand span, the annual salaries of all employees of a large shipping company, the annual salaries of a random sample of 50 CEOs of major companies, the dates of 100 pennies taken from a cash drawer in a convenience store.
Ask students to work in small groups of 3-4 to discuss the video and create a similar question with a choice of items like the problem given by Sal Khan.
3.Whole Class Discussion (10 minutes)
Student groups take turnspresenting their problems. They will first ask the class to think about the solution, then give the solution to the problem and draw the normal distribution for the correct item on the board.
4. Assessment
Students work in groups of 4 to create a problem with at least two variables, make a table of values and create a scatter plot graph to illustrate the data.Groups take turns presenting their problem to the class and explaining how to solve it.
5. Homework assignment for additional independent practice (Note: This homework assignment can be done during a subsequent class period if you have the time.)
Olympics Distance and Time
There are many Olympic events in which participants travel a designated distance as fast as they can. The Competitor who covers the distance in the least time wins the gold medal. Here are nine pages of distance and time data selected from the 2008 Olympics events: swimming, track, rowing, canoe/kayak, and cycling. Complete thisOlympics Distance and Time activity sheetwritten by Bob and George and located on Curriki.
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