Lesson Plan for Variance and Standard Deviation
Week 2: Monday
Course Title/Grade Level: 10th Grade Statistics
Objectives:
 Students will be able to calculate the variance of a population of data by hand and by
using the TI-nspire calculator.
 Students will be able to calculate the standard deviation of a population of data by hand
and by using the TI-nspire calculator.
State Standards:
 CCSS.MATH.CONTENT.HSS.ID.A.2
Use statistics appropriate to the shape of the data distribution to compare center (median, mean)
and spread (interquartile range, standard deviation) of two or more different data sets.
Goals:
 The students will be able to appropriately apply and analyze a population of data.
 The students will be able to accurately use a TI-nspire calculator.
Anticipatory Set/Hook:
 The lesson will start with an activity that requires them all to have TI-nspire calculators. I
will have the students individually measure a piece of string which will be passed around
the room. I will ask them to measure it as accurately as possible, then to secretly record
their data on a scrap piece of paper and put it in an envelope that will be with the string.
Once all of the students have measured the string, I will enter the recorded measurements
in to the calculator and send the list to the students. We will use this data to learn about
variance and standard deviation.
Instructional/Organizational Strategies: Timing of the lesson based on a 50 minute class
 Anticipatory Set: 5-10 minutes
 Lesson: 35-40 minutes
 Exit ticket: 3-5 minutes
Procedure:
 Written in black
Anticipated Student Responses:
 Written in blue.
Direct Instruction:
 Class will begin with the students checking their homework.
 We will review any questions the students have from the homework.
o I have a question on number ##.
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To introduce the lesson I will have the students measure a piece of string and record their
measurements. “Are there any questions before I begin passing the string around?” *(For
more detailed instruction refer to the anticipatory set)
o No.
While the students are measuring the piece of string I will introduce the lesson and
provide an overview of what they will be learning. “Today we will be discussing how to
gain information from a population set of data by looking at the way the data is
distributed. We will use population variance and population standard deviation to help us
gain more information from a set of data than what the mean and median provide us.
Remember last week we found out that some data sets can have the same mean but their
box plots show that their data is spread out much differently. We will be exploring
standard deviation which will provide us with information on how the data is spread.”
Once all data is put in to a table by me, I will send out the table to all of the students.
The students will be instructed to create a histogram of the table they were sent so that
they can have a visual of what the data distribution looks like. Make sure that all of the
histograms have the same bar width and that it is measuring the data by frequency.
Have the students then find the mean of the data by using the calculator. This will be their
first time finding mean on the calculator so the following instructions should be given
*(Note that you should be performing each step with the calculator on the visualizer for
all the students to see the steps you are taking):
o “Everyone must be in the table window for this to work and to be on an blank cell
o First, click on the menu button
o Next, hit the number 3 or scroll down to the word ‘Data’
o Next, hit the number 6 or scroll down to the title ‘List Math’
o From here you hit the number 3 for Mean.
o Or you can scroll down and hit enter once the word is highlighted.
o You should now be back to the table screen with either the word mean in a cell
with parentheses next to it.
o The last step is to hit the ‘var’ button on the calculator and select the title you
gave to the list of measurements.
o Hit enter and a number will pop up in the cell. This number is the calculated
mean.”
“Now that we have found the mean of our data, let us introduce our first vocab term of
the day. Deviation: The distance from each data value to the mean. Symbolically
represented by: xi-𝑥̅ (verbally this reads: data point minus the mean)”
“So I would like each of you to work with your partner to find the deviation of each data
point, and then square each deviation. Once you have all of the deviations squared, add
them up and divide them by the size of the population. In our case the population is our
class size.”
Allow them 5-8 minutes to do all of this, then ask the students what they got for an
answer. If some get the answer wrong, then work with them to figure out where they
made a mistake.
o The student response will depend on the data that is recorded in the activity.
“What all of you have just found is called the variance. Variance is the average of the
2
∑𝑁
̅)
𝑖=1(𝑥𝑖−𝑥
sum of the squares of the deviations of data. Symbolically represented by: s2=
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(Verbally read: s squared is equal to the sum of the deviations squared, divided by the
population while writing it on the board) The Greek symbol∑ , read sigma, means that
you will add all the numbers from i (which equals 1) to N (the size of the population
which is the number of data points we have in our list).”
“Variance is needed to find the standard deviation of our data. Standard deviation is our
last vocab term of the day. Standard Deviation: Measures the spread of data by looking at
how far the data points are from their mean. The importance of finding the standard
deviation will become more apparent in Wednesday’s lesson. Today we are only focusing
on properly finding the standard deviation of a data set”
“Symbolically the standard deviation is represented by s or the square root of s squared;
2
∑𝑁
̅)
𝑖=1(𝑥𝑖−𝑥
s=√𝑠 2 = √
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𝑁
. Now I would like you to find the standard deviation. Round your
answers to the neared hundredth of a decimal. Are there any questions before you get
started? Once you have found the standard deviation, I will briefly explain what
information it tells us about our data set of the measurements you recorded for the piece
of string.”
Allow them time to calculate s. About 30 seconds because they have already found the
variance and standard deviation is only the square root of that.
“What did you find the standard deviation to be?”
o These numbers will depend on the data set that is recorded in the activity.
“Great job everyone! Are there any questions about how we found the standard
deviation?”
“Now, what does standard deviation tell us? If we break it down, the standard deviation
provides us information on how common a range of string measurements are. If we add
our standard deviation to the mean and subtract our standard deviation from the mean we
are given a range that tells us where most of the measurements are located in relation to
the mean. In other words, by finding the standard deviation we are able to find a range
where the most common set of data points will be located. We will go more in depth on
this idea Wednesday.”
“Now let us get more familiar with finding the standard deviation of a set of data by
looking at this next example: Lucy has 20 rose bushes. The number of roses on each bush
is: 9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4. Our task is to find the standard
deviation of the data set.”
“The first step is to find our mean, which is?” (provide them time to calculate the mean)
o 7
“Good, does anyone have a question on how we found the mean?”
o No.
“Now let’s find the variance of the data set. First, we find the deviation for each data
point and square that value. Next, we add up all these numbers. Last, we divide the sum
by the amount in the data set. In this case what is the number we divide by?” Be sure to
work through each of these steps with the students.
o 20
“Good, so what is the variance?”
o 8.9
“Now how do we find the standard deviation once we have the variance?”
o Square root the variance.
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“Great job, any questions so far?”
o No.
“Okay, so the standard deviation is?”
o √8.9 = 2.98 ≈ 3
“Great job. For those of you that did not get the correct answer, what steps did you take?”
o These responses will depend on if any of the students got the standard deviation
wrong.
“We are going to go through one more standard deviation problem, only this time we will
not be calculating the standard deviation by hand because we will be dealing with a much
larger population size. I am sending each of you a file.”
“A class of 530 students is having a penny fundraiser. The first column in the table is a
list of the number of pennies students have collected. The second column is a list of the
number of students that have collected the specified number of pennies. So in the first
row reads: 15 students (frequency column) have collected 130 pennies (penny column)”
“We are looking to find the stand deviation, in this case our job has been made very easy
because it would take too long to calculate this by hand I will show you how to do it
using the calculator.”
o First, you will need to have your cursor in a blank cell.
o Next, hit the menu button and either hit the number 3 or scroll down to data and
hit enter.
o Now, either hit the number 6 or scroll down to list math and hit enter.
o Next, either scroll down to Population Standard Deviation and hit enter or hit
number 9.
o Next, you will need to hit the ‘var’ button on the calculator and select the list
titled ‘peny’
o Last, you will need to type in a comma, then hit the ‘var’ button again and select
the list titled ‘freq’. Hit enter and the number that appears is your standard
deviation.
o Note: if you follow all of the steps again but instead of clicking on Population
Standard Deviation you click on Population Variance and continue with the same
steps I just took you through, then you will be able to calculate the variance.
“So what did everyone get for their standard deviation?”
o The square root of (546550441/265)
“Did anyone get something different??”
o No.
“Now to calculate the decimal form of that you need to have the cell highlighted and hit
the ctrl button then push the letter c. Next, add a calculator page and hit ctrl v and the
square root value should paste in to the calculator page. Last, hit ctrl enter and it will give
you a decimal number. This is how I would like you to answer the homework problems
that are set up with large population sizes. So what did you get as a round answer? Round
the nearest hundredth of a decimal.”
o 88.22
“Great job everyone! Any last questions before you get your assignment?”
“Okay, I have your assignment printed out and I will hand out your exit tickets with the
homework. Remember your exit ticket should have one thing you learned today. You can
either write down an explanation, a definition, or an example with a solution. If you have
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any questions from the lesson, this is the place to write them down and I will answer
them in our next class.”
Assessment:
 I will have them fill out exit tickets and I will be asking questions throughout the lesson
to ensure they understand the material.
Assignments:
 A worksheet that follows this lesson plan.
Multiple Intelligences Addressed:
 Mathematical/Logical: The students will be required to think critically about computing
the variance and standard deviation.
 Intrapersonal: Students will be required to evaluate themselves on how well they
understand the content.
Marzano’s Strategies:
 Note Taking: The students will be required to take notes during the time I lecture.
 Setting Objectives: The objectives for the day will be written on the board and they will
be discussed individually.
 Cues, Questions, and Advance Organizers: I will be asking students questions throughout
the lecture and wait time will be provided such that all students can think about an
answer.
 Homework and Practice: The students will independently practice in class and I will
provide a worksheet for homework.
Accommodations:
 Visual Impairment: Provide enlarged guided notes for the student.
 Hearing Impairment: Speak in a loud, clear voice or wear a microphone. (If the
impairment is severe, then the student should have an interpreter)
 Physical Disabilities (wheelchair): The room will be arranged to allow the student to
properly move about the room.
 Learning Disabilities: Provide guided notes, warm up problems would be daily so a
student with this disability knows what to expect at the beginning of each class, I would
always provide directions on how to organize notes, and I would give them a binder so
that they can place their notes and worksheets in it.
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Name:
Date:
Directions: Round ALL answers to the nearest HUNDRETH of a decimal.
1. There are 20 people in an office. It is recorded that the number of shoes they own is as
follows: 10,12,13,14,15,8,10,5,12,14,10,9,8,14,3,12,15,14,13,9. Find the variance and
standard deviation of this data set.
Number of Shoes
x
Difference from the mean
̅)
(x –𝒙
(Difference from the mean)
̅ )2
(x –𝒙
Sum of (Difference from the mean)
̅)2
∑(x-𝒙
Sum of (Difference from the Mean)2 ÷ by the population size =
Final Step:
Standard Deviation = the square root of the variance
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(i.e. Variance)
2. The following is a frequency table. Calculate the variance and standard deviation based
on the data in the frequency table.
Salary ($)
Frequency
55,000
80
60,000
45
58,000
63
65,000
24
3. You are the Treasurer of your class that consists of 15 students. This year your class is
fundraising for a trip to Chicago. So far the amount of money ($) each student has risen is
as follows. What is the standard deviation of this data set?
100, 200, 150, 30, 100, 125, 135, 160, 180, 165, 150, 150, 190, 200
4. Five hundred pennies were weighed, and the individual penny masses varied from 2.7 g
to 3.4 g. The following table gives the frequencies of each different mass in the set of 500
pennies. Find the mean and the standard deviation of the data.
Mass (g)
2.7
Frequency 2
2.8
15
2.9
57
3.0
111
3.1
138
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3.2
109
3.3
54
3.4
14