Standard deviation lesson

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Topic 11 Lesson 4

Standard Deviation

I CAN…
quantify and analyze the spread of data.

VOCABULARY
• normal distribution
• standard deviation
• variance

Draw

Draw on page 261 in your soft cover

Use page 262 in your softcover for example 1

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Topic 11 Lesson 4

ESSENTIAL QUESTION

Why does the way in which data are spread out matter?

Interpret the Variability of a Data Set

EXAMPLE 1

The makers of a certain brand of light bulbs claim that the average life of the bulb
is 1,200 hours. The life spans, in hours, of a sample of Brand A light bulbs are
shown. How close to the claim were the light bulbs in this sample?

A. What type of data display will provide the best view of the data?

The numbers in the data set span a large range of numbers. To understand how these numbers
relate to the claimed mean, create a histogram.

USE STRUCTURE
How does the shape of the distribution
help you identify the mean?

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Topic 11 Lesson 4

Interpret the Variability of a Data Set

EXAMPLE 1

The makers of a certain brand of light bulbs claim that the average life of the bulb
is 1,200 hours. The life spans, in hours, of a sample of Brand A light bulbs are
shown. How close to the claim were the light bulbs in this sample?

B. What does the display reveal about the data?

The data forms a bell-shaped curve. Data that is in this shape is said to have a normal
distribution.

LOOK FOR RELATIONSHIPS
Most data values in a normally
distributed set are clustered
around the mean. The farther
from the mean, the fewer data
values there are.

Open Ended

What does normal distribution mean?

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Topic 11 Lesson 4

Interpret the Variability of a Data Set

EXAMPLE 1

The makers of a certain brand of light bulbs claim that the average life of the bulb
is 1,200 hours. The life spans, in hours, of a sample of Brand A light bulbs are
shown. How close to the claim were the light bulbs in this sample?

B. What does the display reveal about the data?

When data are normally distributed, the most useful measure of spread is the standard deviation.
Standard deviation is a measure that shows how data vary, or deviate, from the mean.

Open Ended

What is meant by standard deviation? When is it useful?

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Topic 11 Lesson 4

Interpret the Variability of a Data Set

EXAMPLE 1

The makers of a certain brand of light bulbs claim that the average life of the bulb
is 1,200 hours. The life spans, in hours, of a sample of Brand A light bulbs are
shown. How close to the claim were the light bulbs in this sample?

B. What does the display reveal about the data?

In a normal distribution,

•About 68% of data fall within one standard deviation of the mean.

•About 95% of data fall within two standard deviations of the mean.

•About 99.7% of data fall within three standard deviations of the mean.

The standard deviation for this sample of lightbulbs is 75.5 hours.

The mean of this data set is 1,200. Light bulbs with a life span between 1,124.5 hours and 1,275.5
hours lie within one standard deviation of the mean. In a large sample drawn randomly among such
light bulbs, we would always expect about 68% of the life spans to be within one standard deviation of
the mean thanks to the predictive power of the normal distribution.

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Topic 11 Lesson 4

Interpret the Variability of a Data Set

Try It!

1.What is the lifespan of light bulbs that are within 2 standard deviations of the mean?
Within 3 standard deviations of the mean?

EXAMPLE 1

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Topic 11 Lesson 4

Step 2 Find the difference between each data value, x, and the mean, . Then square each difference.

Calculate the Standard Deviation of a Sample

EXAMPLE 2
CONCEPTUAL UNDERSTANDING

The table shows the number of cars sold by an auto sales associate over an eight-
week period. How much variability do the data show?

You can find the variability by solving for the standard deviation for a sample, using the formula

To calculate it, follow these steps.

COMMON ERROR
Remember to square the
differences between each data
value and the mean. Otherwise,
the sum of the differences will
be zero.

Step 1 Find the mean of the data by finding the sum of the data points and dividing by 8. The notation

is used to indicate the mean.

= 17

Use page 263 in your book to take notes
You will need to be able to understand what each part of standard deviation formula means.

Use your calculator to check your work.

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Topic 11 Lesson 4

Calculate the Standard Deviation of a Sample

Try It!

2.The table shows the number of cars sold by the auto sales associate over the next
eight-week period. How much variability do the data show?

EXAMPLE 2

​In your book (page 263) complete each step as shown in the previous example. You should have a table with all calculations.

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Topic 11 Lesson 4

Find Standard Deviation of a Population

The table displays the number of points scored by a
football team during each of their regular season games.

B. The team played in two post-season games, scoring 7 points in one and 14

points in the other. How do these games affect the overall mean and standard
deviation of the number of points scored all season?

EXAMPLE 3

Use graphing technology to find the mean and standard deviation.

Enter all 14 scores as a list in the graphing calculator.

Use the STAT menu, to find the mean and standard deviation.

When the team includes their post-season games, the mean
number of points scored decreases to about 15.9. The standard
deviation remains about 7.3.

Open Ended

What is the difference between δ_x and S_x? Which one will you use for our class?

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Topic 11 Lesson 4

Find Standard Deviation of a Population

Try It!

3.What was the range of points that the team scored in 95% of their regular season
games?

EXAMPLE 3

​Page 263 example 3

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Topic 11 Lesson 4

Compare Data Sets Using Standard Deviation

EXAMPLE 4

The histograms show the life spans of a sample of light bulbs from 2 companies.
The first shows a sample from Brand A and the second from Brand B. The red lines
indicate the mean and each standard deviation from the mean. Compare the
distributions of life spans for the two light bulb brands.

APPLICATION

​Study example 4 on the next two slides. Then answer example 4 on page 263.

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Topic 11 Lesson 4

Compare Data Sets Using Standard Deviation

EXAMPLE 4

The histograms show the life spans of a sample of light bulbs from 2 companies.
The first shows a sample from Brand A and the second from Brand B. The red lines
indicate the mean and each standard deviation from the mean. Compare the
distributions of life spans for the two light bulb brands.

APPLICATION

Both brands of light bulbs have an average life of about 1,200 hours.

The standard deviation for the data set for Brand A is less than the standard deviation for Brand B. This
suggests that Brand A light bulbs show less variability in their life spans.

Since their light bulbs are more consistently closer to the mean, Brand A light bulbs are more
predictable in their life spans than Brand B.

USE STRUCTURE
How does the shape of each graph help
you compare the standard deviations?

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Topic 11 Lesson 4

Compare Data Sets Using Standard Deviation

Try It!

4. Compare Brand C, with mean 1,250 hours and standard deviation 83 hours, to Brands

A and B.

EXAMPLE 4

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Topic 11 Lesson 4

CONCEPT SUMMARY

Standard Deviation

WORDS

Standard deviation is a measure of spread, or variability. It indicates by how much
the values in a data set deviate from the mean. It is the square root of the variance.
The variance is the average of the squared deviations from the mean. When data
are normally distributed (in a bell curve), the mean and the standard deviation
describe the data set completely.

ALGEBRA

Standard Deviation of a Sample

DIAGRAM

Standard Deviation of a Population

Homework:
Page 264 #1-8