Statistical thinking



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STATISTICAL THINKING
Some things to know why it is good to know statistics and some things that will make you better at statistics. Keep these things in mind and you will become more statistically literate. Statistics are all around us and most people are not statistically literate.

DATA ILLUMINATE:


Example: What percent of the American population is black? When white Americans were asked this question the average answer was 23.8%. In fact the census tells us that the true answer is 11.8%. It is interesting to know the correct answer as well as how people think incorrectly.
ALWAYS LOOK AT THE DATA

Graphing data is important. Let’s look at the 2000 presidential election in Florida. After much recounting state officials determined that Bush won over Gore by 537 votes (out of about 6,000,000 total) and hence Bush became president.
What happened in Palm Beach County? The graph called a scatterplot shows votes for Bush and Buchanan. Something remarkable seems to have happened in Palm Beach County. What happened was Palm Beach County used a confusing butterfly ballot that showed candidates on either side with holes to punch in the middle.

It would be easy for a voter who intended to vote for Gore to instead vote for Buchanan. Even if a small percentage made this mistake since so many people voted this could result in hundreds of errors. Buchanan even admitted that most of the people that “voted” for him in Palm Beach County intended to vote for Gore.
What would have happened if we give our best estimate on who people intended to vote for?

It gets more complicated than this. The western panhandle of Florida which favored Bush is in a different time zone and the election was called for Gore 11 minutes before the polls closed. Surely many Bush voters decided not to vote after hearing this. With statistics we could perhaps get a reliable estimate of how many more votes Bush would have gotten. Statistics can’t answer the question of how to decide who is president, but it can be used for an intellectual discussion of who most likely got more votes, who most voters intended to vote for, and who most voters would have voted for if they weren’t given false information before the polls closed!

DATA BEAT ANECDOTES
An anecdote is a striking story that sticks in our minds. It is human nature to pay too much attention to anecdotes and not look at all the data.
What do you think most people will pay more attention to? Consider a story with a mother and her child sick with leukemia that lives very near a huge power line. Imagine the mother describing the buzzing of the power line and blaming on her child’s sickness. On the other hand consider a story of a report in a medical journal on a 5 year 5 million dollar study investigating the relationship between leukemia and power lines. Imagine this study concludes no relationship exits?
To become more statistically literate you need to learn to look at all the data and not focus on a few pieces of data even if they stand out.

BEWARE OF LURKING VARIABLES


A lurking variable is a variable not mentioned that may have a very large impact on the variables mentioned. As an example I am willing to bet that if you measure scores on standardized tests for math and verbal skills for say 4th grade students that are involved in youth soccer and those that are not you would find the students in soccer have higher scores. Does this mean that soccer increases scores on standardized tests for 4th graders? How can this be explained with a variable not mentioned. The variables mentioned are involvement in soccer and test scores.
Take another example. There is a strong relationship between ice cream sales and sunburns. Is there a lurking variable that explains this relationship?

WHERE THE DATA COMES FROM IS IMPORTANT


In a study or an experiment you must be careful about how the data were obtained or produced.

As an example the advice columnist, Ann Landers, once asked her readers the question: “If you had it to do over again, would you have children?” More than 10,000 parents wrote in and 70% said they regret having children. Do you believe that 70% of all parents feel this way? What happened here is that angry parents were much more likely to write in and happy parents were very unlikely to write in. A statistically designed poll would show that the true answer is around 9%. If you are not careful you will conclude 70% when the true answer is 9%!

Here is another example. In 1992 many major medical organizations recommended woman at menopause to take hormone replacement therapy. Studies had shown that women who did this therapy had a 35% to 50% lower chance of heart attack and the risks of the therapy appeared small in comparison. The trouble with studies is that they do not eliminate lurking variables. It is possible just like the parent example above that there is some nonsense in our data. Can you think of a lurking variable that might affect both whether or not the women take hormone replace and also have fewer heart attacks?
Finally experiments were done. In an experiment the women are divided up randomly between the two groups (hormone replacement and placebo). When this was done no difference in heart attacks between the two groups was found. This happened around 2002 and after these experiments were done the National Institutes of Health concluded the conclusions of the earlier studies were wrong and hormone replacement quickly lost its popularity!
Whether a study or an experiment is done you must know how the data were produced to see if the results might be suspect.

VARIATION IS EVERYWHERE


There are a lot of sources of variation. Almost everything varies.
Suppose someone is in charge of a group of salesmen. Certainly this manager wants the sales to go up. The manager can keep track of the sales for each salesman. It would be easy to rate the salesmen (against each other and against themselves over time) by only looking at sales. However, this has its problems. For example there may be many economic factors that would cause sales to go up or down. So just because a salesmen’s sales went down does not mean he is doing a bad job. Also, just by luck one salesman may have a better month than another.

There is another lesson here and that is that it is just about impossible to pinpoint cause and effect in a situation in which there are many variables in play.

Here is another example. Suppose a child in 2nd grade is tested on reading ability on a standardized test at the beginning, middle and end of the school year and the child’s performance is compared with the average of all 2nd graders. The average of all 2nd graders is the bottom graph and the top is the one child. A person that understands the graph, but not how variation works, would say the child lost its lead over the average in the middle and gained it back at the end.






Beg Mid End
The problem is that if you measure the child’s performance at three times it is very unlikely that the three points will line up exactly on a line. So it is about 50-50, just by luck, whether the child will appear to not do so well in the middle or excel in the middle. There are so many places for variation to come in besides how well the child is doing. These include the fact that the test can’t perfectly measure reading ability and the child might have had a bad day. Most likely all that can be ascertained from this graph (there is no scale give) is that the child tends to be a little above the average.
Take as another example two basketball teams playing each other. No matter how you try to control the variables (location, injuries, etc.) if the two teams play over and over the scores will almost surely vary.

CONCLUSIONS ARE NOT CERTAIN


Recall the lottery example at the beginning of the semester. There was a survey of 1523 adults and about 57% had bought a lottery ticket in the last year.

Can we be absolutely sure that more than 50% of all adults have bought a lottery ticket in the last year? What about more than 15%?


Even if we can trust the sampling process there is always some chance that the population percentage is 14% and just by luck we got 57% in our sample.

In the lottery example it did turn out that we were pretty sure it was close to 57%. More precisely, we are 95% sure that the population number is between 54% and 60%. This is assuming we trust the sampling process. But there is always some chance that we are way off.

What about medical experiments on such things as mammograms to help reduce the chance of women dying of breast cancer? What about experiments on drugs that claim to lower cholesterol? Again we can never be absolutely sure. The best we can be is really sure, but we can never be 100% sure.
LIES, DAMN LIES, and STATISICS
Because people have an agenda to push or just not statistically literate there are a lot of bad statistics out there. Many examples will be given in the course. Hopefully you will learn that if statistics is done well then we can learn about what is going on the world. Hopefully you will learn not to be fooled by bad statistics. Or maybe you will even try to fool others with what you learn. The quote “lies, damn lies, and statistics” was popularized by Mark Twain and suggests that damn lies are worse than lies, and statistics are worse than damn lies.
UNFAIR COMPARISONS
Many times in statistics people compare things that should not be compared. Here are three examples:


  1. Suppose in a big city it is found that in all fatal car accidents 25% were under the influence of alcohol and 75% were not. It seems that it is better to be drunk! However, most drivers are not drunk so they make up way more than 75% of drivers. This is a little like comparing deaths in open heart surgery. Surely more than 99% of open-heart surgery deaths occur at hospitals and less than 1% occur at hair salons. But you are not safer having open heart surgery at a hair salon!
  2. Let’s compare percent of children abused in Idaho and Virginia. In Idaho its 22.6% and in Virginia its only 5.9%. Does this mean it is safer for children in Virginia? No, there are vastly different definitions of child abuse from state to state.


  3. How is it that in 1998 North Dakota that was 45th in spending per pupil has a much higher SAT average (by almost 200 points) than New Jersey that was 2nd in spending per pupil? It turns out that most students in North Dakota that take the SAT are going to out of state colleges, so the quality of student taking the SAT is much higher in North Dakota.


STATISTICS
TWO TYPES OF STATISTICS:
DESCRIPTIVE: collecting and presenting data

(Ex: Ask 100 CMU students their favorite pizza place and present the results in a pie chart)


INFERENTIAL: taking data from a sample and making a conclusion about the population

(Ex: Take the results of the above example and say how sure you are that if you interviewed all CMU students the most popular pizza place would be the same as with the 100 students)


Another example of inferential statistics is concluding who will an election before all the precincts have reported. This is a complicated process.
Definitions: Individuals: the objects of interest,

Variable: any characteristic of an individual

Variables can be categorical or nominal (there is no structure, the variables are just place the individuals into categories) or ordinal (there is an order, but the difference between two data does not make sense) or interval (the difference makes sense, but not the ratio) and ratio (differences and ratios make sense). Interval and ratio are considered quantitative. The difference between interval and ratio is that for Interval there is not a starting point of 0.

Categorical examples: state born, area code of phone number, color of eyes

Ordinal examples: grade in a math class, rank in army
Interval example: degrees Fahrenheit
Ratio examples: age, weight, height


PICTURING DATA WITH GRAPHS
Examples of the following will be done in class:
Pie chart: a circle with slices proportional to the different categories of data, if a slice takes up 40% that means that 40% of all the data is in this category
Bar graph: a graph with bars going up (or perhaps to the right) with the heights telling how many (or percent) are in each category.
Note you can’t do a pie chart unless you have all the categories or have an “other” category so you can get percents. You can do a bar graph without knowing all categories or having an “other” category.
All graphs are done to look the best like a piece of art. There are guidelines that give pleasing to look at graphs. For example in a bar graph usually the most popular category comes first, then the 2nd most popular category next and so forth. If there is an “other” category it is usually best to put it at the end. Also bar graphs should have gaps between the categories.
Histogram: On the x-axis is numeric categories and the y-axis is percent or how many in each category. There should be no gaps between the categories.
Note we have done a few histograms in class.
Rough guide for Histograms: 5-15 categories are usually best, the categories should be the same length, not overlapping, and not have any gaps.

What about the following graph? It shows that the under 25 group is by far the worst drivers. Should we outlaw driving for this terrible group of drivers?

Often people are out to mislead you with statistics so be careful, or feel free to use anything you learned in this class to take advantage of people’s lack of statistical literacy!

Stemplots: Often all digits but the last are in the stem; the last digits are the leaves. Again around 5-15 stems are usually best. Stemplots can also be done by making all but the last two digits the stems and the last two digits the leaves, and there is no reason it can’t be more than two also.
To get more stems: split the stems so that digits 0-4 go with the first stem and 5-9 go with the second. This should roughly double the stems.
To get less stems: round the data or just move the break between the stem and leaves one to the left. This should give about 1/10th as many stems.
Make stemplots of the following data sets:
Data set 1: 22, 35, 66, 69, 33, 33, 36, 47, 91, 99, 103, 55
Data set 2: 22.1, 35.4, 66.6, 69.5, 33.3, 33.4, 47.0, 91.0, 99.7, 103.5, 55.6
Data set 3: 22, 35, 46, 49, 33, 33, 36, 47, 41, 49, 53, 55, 38

Timeplots: These have time on the x-axis and the value of some variable on the y-axis

Note that in the histograms, bar graphs, and timeplots, to get the true picture the y-axis should start at 0. Often with timeplots this makes the picture boring, so sometimes we don’t start the y-axis at 0. However, it should be noted that this exaggerates any differences. Again by not starting the y-axis at 0 you can be fooled, or better yet, you can fool others!

Here is an example to show the importance of starting the y-axis at 0:


Gosh, it looks like Albertson’s is by far the lowest and the other’s are ridiculously more! Why does it appear Albertson’s is so much better? Another concern is who decided on the list of groceries, was it an unbiased group or what Albertson’s wanted to include because they knew they had good prices on these items?


Why do graphs such as timeplots, histograms, bar graphs, etc? To make it easier to grasp the properties of the data.





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