Transcript Normal Distribution

Math 2 Unit 4
Standards
 MM2D1. Using sample data, students will
make informal references about population
means and standard deviations.
 A. Pose a question and collect sample
data from at least two different populations
 b. Understand and calculate the means
and standard deviations of sets of data.
Standards (continued)
 c. Use means and standard deviations to
compare data sets
 d. Compare the means and standard
deviations of random samples with the
corresponding population parameters,
including those population parameters for
normal distributions. Observe that the
different sample means vary from one
sample to the next. Observe that the
distribution of the sample means has less
variability than the population distribution.
Essential Questions

How do I calculate the mean and standard deviation and use these measures to
compare data sets?

How does the Empirical Rule apply to a distribution of a data set?

How is the golden ratio related to our study of standard deviation and mean?

How do the sample means vary from one sample to the next?

What happens to the statistics of the data as the sample size approaches the
population size when the population distribution is normal?

How do the statistics of various random samples compare?

As the sample size changes, how do the changes affect the distribution of the data
and, more specifically, the mean and standard deviation?
Central Tendency
Central Tendency: a condition of data
when it clusters or centers around certain
numbers.
Measures of Central Tendency:
 Mean
 Median
 Mode
 Range
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Measures of Variation
Measures of Variation (Dispersion):
 Range
 (Variance)
 Standard deviation
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Practice:
Find the mean, median, variance, and standard deviation
of each data set. Show your work.
OPENING
1.) 6, 22, 4, 15, 14, 8, 8
Mean =
Median =
Variance (σ2) =
Standard Deviation (σ) =
Answers
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1.) 6, 22, 4, 15, 14, 8, 8
Mean = 77 ÷ 7 = 11
Median = 8
Variance = 34
Standard Deviation = 5.831
x
x
Symbols
Symbols:
 s2 = Sample Variance
 s = Sample Standard Deviation
 2 = Population Variance
x
  = Population Standard Deviation
 --x = Mean
The Greek letter  is
pronounced sigma.
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Definitions:
 Sample – a subset of a population with a
manageable size
 Population – the set of individuals, items,
or data from which a statistical sample is
taken
Normal Distribution
 The standard deviation is a good measure of
spread when describing it.
 Many things in life vary normally. (heights of
men).
 Most are around average height, but some are
shorter and some taller.
 The shape will be a bell shape curve. All
normal distributions are bell shaped; however,
all bell shaped curves are not normal.
 If a distribution is a normal distribution, then the
Empirical Rule should apply.
Normal Distribution
Empirical Rule
 If a distribution is normal, then…
 approximately 68% of the data will be located within
one standard deviation symmetric to the mean
 95% of the data will be located within two standard
deviations symmetric to the mean
 99.7% of the data will be located within three
standard deviations symmetric to the mean
Normal Bell Curve
Mean = 23 and standard deviation = 1
Students counted the number of candies in 100 small
packages. They found that the number of candies per
package was normally distributed with a mean of 23
candies per package and a standard deviation of 1
piece of candy.
About how many packages had between 24 and
22 candies?
Draw a normal
curve. Label the
mean and positive
and negative
multiples of the
standard deviation.
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The values of 22 and 24 are 1 standard deviation below
and above the mean, respectively. Therefore, 68% of the
data are located here.
Multiply 100 by 0.68.
Answer: About 68 packages
contained between 22 and
24 pieces.
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Right Skewed Distribution
This is the same as it being called positively skewed.
The tail on the right side is longer.
Left Skewed Distribution
 Also known as a negatively skewed
distribution
 The tail on the left is longer
Determine whether the data {31, 33, 37, 35, 33, 36, 34,
36, 32, 36, 33, 32, 34, 34, 35, 34} appear to be skewed
to the left, skewed to the right, or normally
distributed.
Make a frequency table for the data. Then use the
table to make a histogram.
Value
Frequency
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31
1
32
2
33
3
19
34
4
35
2
36
3
Answer: Since the data are somewhat
symmetric, this is a normal distribution.
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Determine whether the data {7, 5, 6, 7, 8, 4, 6, 8, 7,
6, 6, 4} appear to be skewed to the left, skewed to the
right, or normally distributed.
Answer: negatively skewed
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Standard Terms on TI-83
Stat
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Calc 1-Var Stats
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