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Chapter 2 Review:
Describing Location in a Distribution
Group Members: Stephanie Yooon,
Caroline Lee, Ben McC
The Big Idea
• Describe an individual value’s location within a distribution of data
• Model distributions with density curves
• Z-scores and percentiles provide easily calculated measures of
relative standing for individuals.
• Density curves come in various shapes but all share the property
that the area beneath the curve is 1
• Areas under density curves help estimate proportion of individuals
in a distribution whose values fall in a specified range
• In Normally distributed data, we can use the standard normal curve
• You use this when you meet a new set of data, you can use the
graphical and numerical tools discussed to assess the Normality of
the data
Vocabulary You Need to Know
•
Z-Scores (standardized values): If x is an observation from a distribution that has known mean and standard
deviation, the standardized value of x is: z=x-mean/standard deviation
•
Chebyshev’s Inequality: In any distribution, the percent of observations falling within k standard deviations of the
mean is at least (100) (1-1/k^2)
•
Density Curve: it’s a curve that is always on or above the horizontal axis and has an area exactly 1 underneath it. It
describes the overall pattern of a distribution. Area under the curve and above any interval of values on the
horizontal axis is the proportion of all observations that fall in that interval.
•
The 68-95-99.7 Rule: In a Normal distribution with mean and standard deviation (approx. 68% of the observations
fall within the standard deviation of the mean. Approx. 95% of the observations fall within 2 standard deviations
of the mean. Approx. 99.7% of the observations fall within 3 standard deviations of the mean.)
•
Standard Normal distribution: is the Normal distribution N(0,1) with mean 0 and standard deviation 1.
•
Use of Normal Probability Plots: if the points on a Normal probability plot lie close to a straight line, the plot
indicated that the data are Normal. Systematic deviations from a straight line indicate a non-Normal distribution.
Outliers appear as points that that are far away from the overall pattern of the plot.
•
Median of a density curve: is the “equal-areas point” the point that divides the area under the curve in half
•
Mean of a density curve: is the “balance point” at which the curve would balance if made a solid material
^ Mean and median are the same for a symmetric density curve. They both lie at the center of the curve. The mean of a
skewed curve is pulled away from the median in the direction of the long tail.
Key Topics Covered in this Chapter
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The two principle measurements of the center of a distribution are the mean and
the median
Principle measurements of the spread of a distribution are the range, interquartile
range, variance, and standard deviation.
Principle measurements of position are simple ranking, percentile ranking, and the
z-score.
Adding the same constant to every value in a set adds the same constant to the
mean and median but leaves all the above measures of spread unchanged.
The mean, range, variance, and standard deviation are sensitive to extreme values,
while the median and interquartile range are not.
In skewed left data, mean is usually less than the median, while in skewed right
data, the mean is usually greater than the median
Boxplots visually show the five-number summary: the minimum value, the first
quartile, the median, the third quartile, and the maximum value, and usually
indicate outliers as distinct points.
NOTE: Two sets can have the same five-number summary and thus the same
boxplots but have dramatically different distributions.
Formulas You Should Know
• Finding the z-score:
• Chebyshev’s Inequality:
Calculator Key Strokes
Normal probability plots on TI-83/84
1.
Press Stat
2.
Choose Calc
3.
Choose 1:1 Var stats
4.
Press 2nd 1 (L1)
5.
Go to plot 1
6.
Turn on plot data with x values on horizontal axis
7.
Use ZoomStat
Finding areas with shadenorm on TI-83/84
1.
Press 2nd VARS (DISTR), then choose DRAW and 1: ShadeNorm
2.
Complete the command ShadeNorm (125,1E99,100,15) and press Enter
Finding areas with normalcdf TI-83/84
1.
Press 2nd VARS (DISTR) and choose 2: normalcdf
2.
Complete the command normalcdf (125, 1E99,100,15) and press Enter
Finding values with invNorm TI-83/84
1.
Press 2nd VARS (DISTR) then choose 3: invNorm
2.
Complete the command InvNorm (.9, 100, 15) and press Enter. Compare this with the command
invNorm (.9)
Example Problem
• Which of the following statements are true?
I.
If the right and left sides of a histogram are mirror images of each
other, the distribution is symmetric
II. A distribution spread far to the right side is said to be skewed to
the right
III. If a distribution is skewed to the right, its mean is often greater
than its median.
A) I only
B) I and II
C) I and III
D) II and III
E) None of the above gives the complete set of true responses
Helpful Hints
• Remember how to use your z score chart
• Z score is up the point, so everything left of
the value is the z score
• Don’t forget your calculator is your best friend
(:
• This presentation is credible. Use it!
• AP STATISTICS IS THE BEST REMEMBER TO
ENJOY IT <3