Name ______
AP STATISTICS CHAPTER 1:
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1. What is the difference between descriptive and inferential statistics?
2. How is a sample related to a population?
3. What are two types of numerical variables?
4. What is the difference between a categorical and a quantitative variable?
5. Define Distribution:
6. What are 3 attributes of a good chart?
The following terms help describe the shape of a distribution? Draw a sketch to help explain each term.
BIMODAL VS UNIMODAL –
UNIFORM DISTRIBUTION -
SYMMETRIC DISTRIBUTION –
SKEWED LEFT –
SKEWED RIGHT –
APPROXIMATELY NORMAL -
7. What acronym should you use to describe the overall pattern of a distribution?
8. How are histograms and stemplots similar? How are they different?
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The following stem and leaf plot displays the number of home runs hit, per season, by Barry Bonds from his rookie year in 1986 to the year 2003.
1 / 6 92 / 4 5 5
3 / 3 3 4 4 7 7
4 / 0 2 5 6 6 9
5
6
7 / 3
MEASURING SPREAD: can be used to describe the spread of a distribution.
The 5-number summary: Boxplot:
The inter-quartile range (IQR):
Call an observation an if it falls more than above Q3 or below Q1.
Does the Barry Bonds data contain any outliers?
MEASURING CENTER: We use the following measures to describe the center of a distribution:
MEAN –
MEDIAN –
What are the mean and median of the Barry Bonds data?
What is the effect on the mean and the median if the outlier (73) is changed to 51?
A statistic is said to be if it is unaffected by changes to individual data points.
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The most common statistic used for describing spread is standard deviation
Sample variance: s2 =
Sample standard deviation: s =
Are the variance and standard deviation resistant?
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x / Deviations/ Squared deviations
16
19
24
25
25
33…
Why divide by n-1 and not n?
"Some questions are more easily asked than answered" Edward Sapir(?)
Possibly the most frequently asked and least frequently answered question is why does the definition of the standard deviation involve division by n-1, when n might seem the obvious choice. This is a question which perplexes introductory statistics students and calculator manufacturers alike. The explanations given in calculator manuals tend to range from obscure to fanciful, and both options are given on the keypad, usually labelled sn-1 and sn to add to the confusion. (The symbol s is reserved for the standard deviation of a random variable or a population/distribution, an entity which lecturers valiantly try but usually fail to keep distinct from its sample counterpart.) Australian students first meet the standard deviation in secondary school, where the definition given does indeed involve division by n. This definition is preferred to avoid the question about the n-1 being raised it would seem. Secondary school teachers have a hard enough life as it is. And it must be remembered that the difference between the two definitions is largely academic for all but the smallest of sample sizes (say, less than 10 observations). So, don't get too agitated by the revamped definition. The truth can be told but the telling usually quells the desire to know. If the fire still burns in your belly, read on.
The most widely accepted explanation involves the concept of unbiasedness, and I see some have stopped reading already. If you fire arrows at a target and consistently hit a mark 5 cm to the left of the bullseye, there is something wrong with your aim. It shows a bias. The definition of the sample variance which involves division by n has this flaw. It consistently underestimates the variance of the population/distribution from which the sample was drawn. The n-1 formula fixes the astigmatism. Compelling and relatively simple as this argument is, it doesn't quite ring true. Both definitions of the sample standard deviation produce biased estimates of the standard deviation of the population/distribution, although the n-1 alternative is less biased. If you're after unbiasedness, why not use a definition which gives you unbiasedness where it's needed - on the original scale of measurement, rather than on the squared scale. Such a contender exists, but it involves gamma functions in the definition, and I see quite a few more people have drifted away. (Gamma functions extend the concept of factorials to non-integers.)
The real reason is a simple housekeeping issue. If you deal with the n-1 straight away in the definition of the standard deviation, it doesn't keep popping up in every subsequent procedure involving the standard deviation, to the increasing annoyance of all concerned. The subsequent procedures in question involve the definition of the t and c2 distributions where the issue of degrees of freedom arises. Degrees of freedom means what it says - in how many independent directions can you move at once. If you're a point moving on a page, you are moving in two dimensions and you have correspondingly two degrees of freedom. The freedom to move up the page and the freedom to move across it. Any motion on a page can be described in terms of these two independent motions. Now consider a sample of size n. It inhabits an n dimensional space. There are n degrees of freedom in total. Each sample member is free to take any value it likes, independently from all the others. If however, you fix the sample mean, then the sample values are constrained to have a fixed sum. You can let n-1 of them roam free, but the value of the remaining sample value is determined by the fixed sum. The space inhabited by the deviations from the sample mean is thus n-1 dimensional rather than n dimensional, since the deviations must sum to zero. The sum of the squared deviations, although looking like a sum of n things is actually a sum of only n-1 independent things, and its natural divisor - its degrees of freedom - is also n-1.
But wait, there's more. Degrees of freedom will return to haunt you if and when you do analysis of variance (ANOVA to its friends). You will be ahead of the game if you grasp the concept now. Degrees of freedom can neither be created nor destroyed. You start off with n, the sample size. You use up a few trying to estimate the structure of the mean. For example, the mean could be a straight line, as in simple linear regression. You need two degrees of freedom to estimate the two characteristics possessed by all straight lines - a slope and an intercept. These characteristics are called parameters. So, two degrees of freedom have gone into the mean. This is the signal. Everything else in this model is noise. The remaining n-2 degrees of freedom go into estimating the one parameter which describes the noise - the variance. If you're not part of the solution (the signal or mean) you're part of the problem (the noise). The simplest model is the one which says the mean is a single constant, ably estimated by the sample mean. Everything else is just inexplicable variation about that constant. That's n-1 degrees of freedom's worth of noise, all kindly donated to the sample variance.
FROM: http://www.maths.murdoch.edu.au/units/statsnotes/samplestats/stdevmore.html
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In problem 1.14 on page 23, the salaries of CEO’s from 59 small businesses was provided (in thousands of dollars). This data is now held safely in your calculators, in L1. Give the mean and standard deviation of this data, and create a boxplot. Describe the shape of the distribution.
Consider the following changes to the data. Use your graphing calculator to again determine the mean and standard deviation, and describe the shape of the distribution. Create side-by-side boxplots to display the distribution.
1. The salary of each CEO is increased by 80 thousand dollars. (Use L2)
2. The salary of each CEO rises 30%. (Use L3)
3. The salary of each CEO falls 40%, but then rises $20,000. (Use L4)
How do changes to a data set, through addition, subtraction, or multiplication, affect the mean and standard deviation of a distribution?
LINEAR TRANSFORMATION: A linear transformation changes the original variable x into the new variable xnew given an equation by the form: xnew = a + bx .
Effects of a:
Effects of b:
Effects on the shape of the distribution:
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Consider the following data and graph concerning heights of freshmen boys:
What percent of freshmen have heights below 70 inches?
What is the median height of a freshman boy?
DEFINITION: The pth percentile of a distribution is the value such that p percent of the observations fall at or below it.
What is the 30th percentile of this distribution?
What is the 90th percentile?
At what percentile would a student of height 71 inches fall?
An OGIVE, or a relative frequency graph allows us to view the relative standing of individual observations. The y-axis in this type of graph tells the relative cumulative frequency for the distribution.
Create an ogive for the height of freshmen males data set.
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SUMMARY/QUESTIONS TO ASK IN CLASS