Download S13 Examples and Activities2 - mi-math

HSCE: S1.3.1 Explain the concept of distribution and the relationship
between summary statistics for a data set and parameters of a
distribution.
Clarifying Examples and Activities:
Types of Distributions:
 Stemplots, preferable for data sets with small range
 Dotplots and number line plots, original data points are preserved
 Histograms, preferable for larger data sets,
the original data are usually lost; the choice of bin width
determines how histogram displays the data
 A boxplot also gives a picture of the symmetry of a dataset,
and shows outliers very clearly. most useful when comparing
two or more sets of sample data.
Shape of Distributions:
 Shape is commonly categorized as symmetric, left-skewed or
right-skewed, and as uni-modal, bi-modal or multi-modal.
Clustering implies that the data tends to bunch up around
certain values. The shape of a dataset will be a main factor in
determining which set of summary statistics best summarizes
the dataset.
Summary Statistics of Distributions:
 Measures of center are mean and median.
 Measures of spread of data are range, mean deviation,
interquartile range, and standard deviation.
Websites:
Educator’s data sources:
http://www.wnba.com/
http://www.nba.com/
http://www.nascar.com/
http://www.sportingnews.com/
http://www.keypress.com/x2814.xml
http://exploringdata.cqu.edu.au/
http://lib.stat.cmu.edu/DASL/
Histogram Applet
This applet is designed to teach students how bin widths (or the
number of bins) affect a histogram.
http://www.stat.sc.edu/~west/javahtml/Histogram.html
Example 1:
Choose a data set that is normally distributed; Students will construct a dot plot
of the data. Calculate the average of the data; draw a vertical line to indicate the
location of the mean on the dot plot. Discuss the “picture plot” that results from
the data and sketch a bell shaped curve over the plot. Explain that the dots are a
picture of the actual data while the bell shaped curve is the educated guess
about the distribution of the entire population. Also notice that the mean and
median are centered on the plot.
Explore the concept of average deviation from the mean being “ the average of
the distances of each data point from the mean.” How would the average
deviation from the mean vary if the distribution was skewed, bi-modal, or uniform
shaped? Explain.
HSCE: S1.3.2 Describe characteristics of the normal distribution, including
its shape and the relationships among its mean, median, and
mode.
Clarifying Examples and Activities:
Example 1:
Given the following normal curve, identify the key characteristics regarding
its shape, and compare relationships among mean, median and mode.
In the first curve, shade the area that encompasses one standard deviation
above and below the mean.
In the second curve, shade the area that encompasses two standard
deviations above and below the mean.
In the third curve, shade the area that encompasses three standard
deviations above and below the mean.
Answer the following questions:
1. What percent of the total area under the curve is represented by the
shading in the first graph?
2. What percent of the total area under the curve is represented by the
shading in the second graph?
3. What percent of the total area under the curve is represented by the
shading in the third graph?
HSCE: S1.3.3 Know and use the fact that about 68%, 95%, and 99.7% of
the data lie within one, two, and three standard deviations of
the mean, respectively in a normal distribution.
Clarifying Examples and Activities:
Example 1:
Given the following normal curve, identify the key characteristics regarding
its shape, and compare relationships among mean, median and mode.
In the first curve, shade the area that encompasses one standard deviation
above and below the mean.
In the second curve, shade the area that encompasses two standard
deviations above and below the mean.
In the third curve, shade the area that encompasses three standard
deviations above and below the mean.
Answer the following questions:
4. What percent of the total area under the curve is represented by the
shading in the first graph?
5. What percent of the total area under the curve is represented by the
shading in the second graph?
6. What percent of the total area under the curve is represented by the
shading in the third graph?
HSCE: S1.3.4 Calculate z-scores, use z-scores to recognize outliers, and use
z-scores to make informed decisions.
Clarifying Examples and Activities:
Example 1:
Collect data on a topic like IQ, ACT or SAT scores (known to be normally
distributed). Then give students a list of hypothetical values and have them
calculate the z scores using the formula z = (x- )/. Use diagrams to show
whether a value is likely to have a score that is negative vs. positive (below
or above the mean) or is likely to be in the tail (beyond 3) – a reference to
the empirical rule S.1.3.3 would be good.
Example 2:
John took a math exam and earned a score of 72. The class results were
normally distributed with a mean score of 82 and a standard deviation of 4.
Which of the following is a true statement?
a. John’s score could be considered an outlier.
b. John’s z score is positive.
c. John did better than at least 5% of the class.
d. None of the above
Example 3:
Elissa earned a score of 400 on her final exam. The class results were
normally distributed with a mean of 350 and a standard deviation of 30.
Which of the following is Elissa’s z-score?
e.
f.
g.
h.
-1.67
-.6
.6
1.67