Unit 3 Lesson 1 (4.1) Numerical Methods for Describing Data 4.1: Describing the Center of a Data...
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Transcript of Unit 3 Lesson 1 (4.1) Numerical Methods for Describing Data 4.1: Describing the Center of a Data...
![Page 1: Unit 3 Lesson 1 (4.1) Numerical Methods for Describing Data 4.1: Describing the Center of a Data Set.](https://reader035.fdocuments.net/reader035/viewer/2022070400/56649f135503460f94c27cb2/html5/thumbnails/1.jpg)
Unit 3 Lesson 1(4.1)
Numerical Methods for Describing Data4.1: Describing the Center of a Data Set
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60% of the sample was satisfied with their cell phone service.
6.0159ˆ p
What values are used to describe categorical data?Suppose that each person in a sample of 15 cell phone users is asked if he or she is satisfied with the cell phone service.
Here are the responses:Y N Y Y Y N N Y Y N Y Y Y N N
What would be the possible responses?
Find the sample proportion of the people who answered “yes”:
Pronounced p-hatThe population proportion is
denoted by the letter p.
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Population Parameter -
• Fixed value about a population• Typically unknown
Suppose we want to know the MEAN length of all the fish in Lake Lewisville . . .
Is this a value that is known?
Can we find it out?
At any given point in time,
how many values are
there for the mean length of fish in the
lake?
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Sample Statistic -• Value calculated from a
sample
Suppose we want to know the MEAN length of all the fish in Lake Lewisville.
What can we do to estimate this unknown population characteristic?
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Measures of Central TendencyMedian - the middle value of the data;
it divides the observations in half
To find: list the observations in numerical order
even is if values middle two the of average
odd is is value middle single median sample
n
n
Where n = sample size
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Suppose we catch a sample of 5 fish from the lake. The lengths of the fish (in inches) are listed below. Find the median length of fish.
3 4 5 8 10
The numbers are in order & n is odd – so
find the middle observation.
The median length of fish is 5 inches.
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Suppose we caught a sample of 6 fish from the lake. The median length is …
3 4 5 6 8 10
The numbers are in order & n is even – so find the middle two observations.
The median length is 5.5
inches.
Now, average these two values.
5.5
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Measures of Central TendencyMean is the arithmetic average.
–Use m to represent a population mean
–Use x to represent a sample mean
Formula:
S is the capital Greek letter sigma – it means to sum the values that
follow
Population Parameter
Sample Statistic
m is the lower case Greek letter mu
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Suppose we caught a sample of 6 fish from the lake. Find the mean length of the fish.
3 4 5 6 8 10
To find the mean length of fish - add the observations and divide
by n.
61086543
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What happens to the median & mean if the length of 10 inches was 15 inches?
3 4 5 6 8 15
The median is . . .
5.5
The mean is . . .
61586543
6.833
What happened?
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What happens to the median & mean if the 15 inches was 20?
3 4 5 6 8 20
The median is . . .
5.5
The mean is . . .
62086542
7.667
What happened?
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Some statistics that are not affected by extreme values . . .
Is the median affected by extreme values?
Is the mean affected by extreme values?
NO
YES
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Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width of 1.)
Mean =Median =
3 5 6 10 6 7 7 8 4 5 6 4 7 5 9 9 8 7 6 8
6.5
Calculate the mean and median.
6.5
Look at the placement of the mean and median in this symmetrical distribution.
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Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width 1.)
Mean =Median =5.5
6.8
Calculate the mean and median.Look at the placement of the
mean and median in this skewed distribution.
3 5 6 10 15 7 3 3 4 5 6 4 12 5 3 4 8 13 11 9
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Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width of 1.)
Mean =Median =
8.57.75
Calculate the mean and median.Look at the placement of the
mean and median in this skewed distribution.
3 5 6 10 10 7 10 8 9 5 6 4 9 10 9 9 10 7 10 8
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Recap:• In a symmetrical distribution, the mean and
median are equal.• In a skewed distribution, the mean is pulled
in the direction of the skewness.
• In a symmetrical distribution, you should report the mean!
• In a skewed distribution, the median should be reported as the measure of center!
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Summary
p = Population Proportion (parameter)
= Sample Proportion (statistic) = Population Mean (parameter) = Sample Mean (statistic)When describing center…
Measure of CenterSymmetric MeanSkewed Median
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Homework
• Pg.110:
#4.1, 4.2ab, 4.3, 4.14