Download Calculating the Standard Deviation – Lesson Plan

Calculating the Standard Deviation – Lesson Plan
Class:
Date:
Standard(s):
Students compute the variance and the standard deviation of a distribution of data.
Key Objective(s):
Here you'll learn the meaning of standard deviation, understand the percents associated with
standard deviation, and calculate the standard deviation for a normally distributed random
variable.
Learning Activities and Timing:
Exit Criteria:
When students finish this lesson, they should be able to calculate the standard deviation of both
a population and of a sample for a normally distributed random variable. Students should also
understand and be able to apply the percentages associated with normal distribution to solve
problems.
Teaching Strategies and Tips:
In working through the lesson on Calculating the Standard Deviation, there are several learning
objectives you want the students to know. You also want students to be able to demonstrate
their knowledge of these concepts by the end of the lesson.
1. Students need to understand the difference between a population and a sample. A
population is the total of all subjects, possessing common characteristics, which are being
surveyed or studied. A sample is a group of subjects selected from a population. The
formula used to calculate the standard deviation of a data set depends upon whether the
data set represents a population or a sample.
2. Students must know the steps involved in calculating the standard deviation of a given data
set.



  is
The first step is to determine the mean x or  of the data values. The symbol x
used for the mean of a sample while the symbol    is used for the mean of a
population. Whether the data set is representative of a population or a sample, the
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mean of each is calculated in the same manner – divide the sum of the data values by
the number of data values.

The second step is to find the difference between the mean and each data value.
 x  x  or    x
i
i



2
or    x i  .
2
values.
2

The third step is to square each difference found in step 2. x  x i

The fourth step is to determine the sum of the x  x i

The fifth step is to divide this sum by the number of data values (for a population) or by
the number of data values less one (for a sample).

The final step is to take the square root of the quotient obtained in step 5. In other
words, find the square root of the variance.

The following formulas represent the steps necessary to calculate the standard deviation
of either a sample or of a population.
s 


 x  xi
n 1

 or    x 
2
2
i
  xi 
2
is for the standard deviation of a sample and  
n
is for the
standard deviation of a population.
3. Students need to know and understand the 68-95-99.7 rule for applying percentages to the
normal distribution curve.
A survey was conducted at a local high school to determine the number of hours that a student
studied for the final Math 10 exam. To achieve a normal distribution, 325 students were
surveyed. The results showed that the mean number of hours spent studying was 4.6 hours
with a standard deviation of 1.2 hours.
a.
b.
c.
d.
Draw a normal curve showing all the values.
How many students studied between 2.2 hours and 7 hours?
What percentage of the students studied for more than 5.8 hours?
Harry noticed that he scored a mark of 60 on the Math 10 exam but had studied
for ½ hour. Is Harry a typical student? Explain.
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<Figure 6.03.01>
a)
68%
95%
99.7%
2.2
1.0
4.6
3.4
7.0
5.8
8.2
b) 95% of students = 0.95 × 325 students = 308 students
Therefore 308 students studied between 2.2 and 7 hours.
c) ½ (99.7 % - 68 %) = ½ (31.7 %)
= 15.85 %
15.85 % of the students studied longer than 5.8 hours.
d) Harry is not a typical student. The mean is 4.6 hours; therefore the majority of students
studied more than 4 hours more than Harry did for the exam. Harry is lucky to have received a
60% on the exam.
Common Errors:

Forgetting the Formula and/or misunderstanding the symbols. Students often
forget the formula that must be used to calculate the standard deviation of the given data
set. In addition, they often forget what is represented by the symbols in the formula. The
formula and an explanation of its symbols should be clearly posted in the classroom for
reference for the students to use when solving problems involving the calculation of the
standard deviation of a set of numerical data.

Errors in calculations. When solving problems requiring the students to determine the
standard deviation of a data set, errors are often made in either the addition of the data
values or in the division of the sum by the number of data values. To avoid these
computational errors, students should be encouraged to use a calculator. Students
frequently use incorrect percentages when solving problems dealing with normal
distribution. A poster displaying a normal distribution curve and the percentages associated
with each standard deviation should be made available for student reference when solving
such problems.
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Differentiated Instruction:

Intrapersonal Learners: Students, who benefit by reflecting on what they have learned,
should be encouraged to keep a record of what they have learned in their math journal. As
they advance further into the world of statistics, the records in their math journal will serve
as a reference to review every topic they have explored.

Visual Spatial: Students who have the visual-spatial learning style will learn the concepts
presented here by watching the videos associated with this lesson.
Enrichment:
95% of all cultivated strawberry plants grow to a mean height of 11.4 cm with a standard
deviation of 0.25 cm.
a) If the growth of the strawberry plant is a normal distribution, draw a normal curve
showing all the values.
b) If 225 plants in the greenhouse have a height between 11.15 cm and 11.65 cm, how
many plants were in the greenhouse?
c) How many plants in the greenhouse would we expect to be shorter than 10.9 cm?
Problem Solving:
As Tomato Inspector 007, my job was to ensure that bulk tomatoes sold at Sams did not vary
too much in their diameters. A variation of up to 0.9 was acceptable. I randomly selected 12
tomatoes and measured their diameters. The following is a list of the diameters of the
tomatoes:
6.1, 6.4, 6.5, 6.5, 6.6, 6.9, 7.3, 8, 8.1, 8.1, 8.3, 8.5
Should I make the produce manager take these tomatoes out of the display? Justify your
answer.
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