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Chapter 2: Modeling Distributions Of Data
Key Vocabulary:
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density curve
 mu
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 sigma
outcomes
simulation
normal curve
normal distribution
Calculator Skills:
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inflection point
68-95-99.7 rule
percentile
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N  , 
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standardized value
z-scores
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randInt
X[35, 185]25
Y[-.01, .02].01
rand
ShadeNorm(lowerbound,
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standard normal
distribution
normal probability plot
normalcdf(lowerbound,
EE (1E99 and -1E99)
2.1 Measures of Relative Standing and Density Curves
1. What is a percentile? The nth percentile of a distribution is the value with p
percent of the observations less than it. (Recall Q1, Q2, and Q3.)
2. Is there a difference between the 80th percentile and the top 80%? Explain.
Yes – the 80th percentile is the point where 80% of the data values fall
below. The top 80% are the observations represented when 20% of the
low scores are removed (i.e. the percentile range from 20 – 100% remains)
Reading Guide
Chapter 2: Modeling Distributions Of Data
3. Is there a difference between the 80th percentile and the lower 80%? Explain.
No – the 80th percentile is the values in the distribution up to 80% or it is
the value that has 80% of the observations below it. The lower 80% refers
to the same values.
4. What is the difference between relative frequency and cumulative relative
frequency? Relative frequency is the ratio of how often each observation
occurs expressed as a percent; cumulative relative frequency is a running total
of the relative frequencies
5. What is an ogive plot? How is it constructed? An ogive is a cumulative
frequency graph. It is used to describe the position of an individual within a
distribution or to locate a specified percentile of the distribution. It is
constructed/graphed using a percentage range from 0 to 100 on the y axis
and plotting the distribution of the variable’s corresponding cumulative
relative frequencies on the x axis.
6. Explain how to standardize a variable. To standardize a value, subtract the
mean of the distribution from the value in question and then divide by the
standard deviation. A standardized value is also known as the z-score.
x - mean
=
STDEV
z
7. What is the purpose of standardizing a variable? To find out how many
standard deviations from the mean an observation falls and in what
direction. You can also use z scores to compare the position of individuals in
different distributions as well as to express variables on a common scale.
Reading Guide
Chapter 2: Modeling Distributions Of Data
8. What effect does standardizing the values have on the distribution? It allows
the values in different distributions to be expressed on a common scale;
converting observations from their original unit to a standardized
(common) scale.
9. What is a density curve? A graphical display of a set of data that describes its
overall pattern. It is always on or above the horizontal (y) axis, and has an
area of exactly 1 underneath it.
10. What does the area under a density curve represent? The area under the
curve and above any interval of values on the y-axis is the proportion of all
observations that fall in that interval. Basically, areas under a density curve
represent proportions of the total number of observations.
11.
Where is the median of a density curve located? The median, also known
as the “equal-areas point” of a distribution, is the point on a symmetric
density curve that has half the area under the curve to its left and the
remaining half of the area under the curve to its right. In other words, the
median is the point that divides the area under the curve in half.
NOTE: It’s not so easy to graph or depict the median on a skewed
density curve.
12. Where is the mean of a density curve located? The mean, also known as the
“balance point” of a distribution, is located at the point where the curve
would balance if made of solid material.
13. Where is the mean in relation to the median on a density curve that is
symmetric? Draw a picture.
See p. 102; Figure 2.9 a
Reading Guide
Chapter 2: Modeling Distributions Of Data
14. Where is the mean in relation to the median on a density curve that is… (Draw
a picture)
symmetric? Directly in the middle of the curve – see p.102 Figure 2.9a
Skewed right? Not in the middle but pulled away from the middle to the
right – see p. 102 Figure 2.9 b
Skewed left? Not in the middle but pulled away from the middle to the left
– graph is opposite of skewed right.
15. What is the difference between x and m and sx and s ? Both represent the
mean and standard deviation. However, X bar and Sx are the means and
standard deviations computed from actual observations, i.e. samples of a
population. m and s are ideal descriptions of data and refers to the entire
population vs. a sample.
16. Describe the shape of the normal density curve. Smooth, balanced,
symmetric, above horizontal axis
2.2 Normal Distributions
1. How would you describe the shape of a normal curve? Draw several
examples.
2. Where on the normal curve are the inflection points located?
Reading Guide
Chapter 2: Modeling Distributions Of Data
3. Explain the 68-95-99.7 Rule.
4. What is the standard normal distribution?
5. What information does the standard normal table give?
6. How do you use the standard normal table (Table A) to find the area under
the standard normal curve to the left of a given z-value? Draw a sketch.
7. How do you use Table A to find the area under the standard normal curve to
the right of a given z-value? Draw a sketch.
8. How do you use Table A to find the area under the standard normal curve
between two given z-values? Draw a sketch.
Reading Guide
Chapter 2: Modeling Distributions Of Data
9. How do you use Table A to find x when you know the percent or area under
the curve?
10. Describe two methods for assessing whether or not a distribution is
approximately normal.
11. How can you produce a normal probability plot on a TI 83/84+, and what
should this look like if the data are normal?
12. What will the normal probability plot look like if the distribution is skewed?
13. What information needs to used when using “ShadeNorm(” and what result
will the calculator give?
14. What information needs to used when using “normalcdf(” and what result
will the calculator give?
Reading Guide
Chapter 2: Modeling Distributions Of Data
15. What information needs to used when using “invNorm(” and what result
will the calculator give?
Reading Guide