Statistical Reasoning

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Statistical Reasoning

• For Communication Majors

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Mean

• This is a common statistic, and its simple.
• When we refer to the average, this is usually
  what we mean.
• Add the values and divide by the number of values
  you have.

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Mean Example

• A weekly newspaper has seven employees. Whats
  the mean salary? Here are their salaries
• Editor -- 37,000
• Assistant Editor -- 32,000
• Reporter -- 28,000
• Ad Sales Manager -- 38,000
• Ad Sales Agent -- 31,000
• 2 Circulation People -- 22,000 each

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Mean Example

• Calculation Add 37,000 32,000 28,000
  38,000 31,000 22,000 22,000 210,000. Then
  divide by 7 Mean salary is 30,000
• NOTE Mean can be deceptive if there is a wide
  spread in the numbers. For example, if the editor
  and ad sales manager made 60,000 each, the sales
  agent made 40,000, and each of the other workers
  made 12,500, the mean would be the same, but the
  picture of the average salary at the newspaper
  would be much different.

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Median

• The median means the middle.
• It is the value in the dead center of the list of
  values when they are lined up from largest to
  smallest.
• It represents the average person or group. For
  example, if we say the average household or
  the average worker, then what we are looking
  for is the median, as in ordinary or most
  common. We arent really talking about the
  average or mean.

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Median Example

• Consider the newsroom salaries used in the
  previous example lined up from largest to
  smallest 38,000, 37,000, 32,000, 31,000, 28,000,
  22,000, 22,000.
• The salary in the middle, the median, is
  31,000.
• If the halfway lies between two numbers, split
  them.

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Percent Change

• If the city increased parking fines from 10 to
  15, by what percentage did the fines increase?
• This is simple, too. Subtract the old value from
  the new value (15-105), then divide by the old
  value (5/100.5). Multiply the result by 100
  (0.5x100 50 percent ) and thats the percent
  change.
• 15-105 5/100.5 0.5x100 50 percent.

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Tax Example

• If the average property tax increased by 2,000 a
  year (Were using median here to find 2,000),
  what is the average percent change?
• New value 10,000
• Old value 8,000
• 10,000 8,000 2,000
• 2,000/8,000 .25
• 100x.25 25 percent
• So the percent change is 25 percent

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Per capita, Rates and Comparisons

• Per capita refers to the rate per person. It
  helps make comparisons among large groups, like
  cities.
• To get per capita, simply divide the number of
  incidents by the number of people.
• A Southern city with a population of 450,000
  experienced 16 murders during 2009. What is the
  citys murder rate per 100,000 population?
• 450,000/100,000 4.5 16/4.5 3.5 per 100,000

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Per capita example

• If a city has a population of 600,000 and
  experiences 12 murders a year, the per capita
  murder rate would be 12 divided by 600,000.
• To avoid tiny decimals, divide 600,000 by 100,000
  and report the rate as a number per 100,000
  population.
• 600,000/100,000 6 12/6 2, so the murder rate
  is 2 per 100,000 people.
• You can also find the percent change of the per
  capita rate over time to discover the trend in
  the murder rate.

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Comparison Example

• Suppose you want to know how dangerous the city
  is compared to other cities. Our example city has
  a population of 600,000 with 12 murders. A nearby
  city has 26,000 and 4 murders. Which is more
  dangerous? Find the per capita murder rate of
  each to know.
• Per capita rate for City 1 is 2 per 100,000 per
  capita rate for City 2 is 4 per 26,000. City 2 is
  more dangerous because it has 15.4 murders per
  100,000 (4/.26 15.38) people compared to City
  1s 2 murders per 100,000.

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Standard Deviation

• In most situations, most people or values will
  group toward the middle.
• Those that dont are different.
• If many group outside the middle, then that tells
  you something about the situation it tells you
  that whatever youre looking at isnt expected.

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Standard Deviation

• For normal situations, the curve will look
  bell-shaped, like this

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Standard Deviation

• Most healthy women will eat between 1,700 and
  2,000 calories a day. If you plot how many
  calories women eat, each womans intake will be
  one value. Plot them on a sheet of paper along a
  line and most of the values (number of calories)
  will land in the middle of the spread. That will
  be what is called a normal distribution.

Normal distribution
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Standard Deviation

• In a normal distribution, about 68 of the women
  will gather in the middle. They are one standard
  deviation away from the middle on either side.
  (The blue area on the graph.)

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• Two standard deviations away will account for
  about 95. (The blue areas and the brown areas.)
• So, 95 of the values in most situations will be
  considered normal. However, all but the middle
  68 will be somewhat abnormal, but not
  excessively abnormal.

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• Three standard deviations away from the middle
  will account for about 99 of the values. (The
  blue, brown, and green areas). The values in the
  green areas are more abnormal, but we expect
  about 4 of values to fall into these areas,
  because life is not perfect.

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Standard Deviation

• If a scientific study concludes that 99 of the
  values fall within three standard deviations,
  then you have a normal situation and the
  conclusions can be trusted.
• A good public opinion survey, for example, that
  concludes Americans support the Presidents
  policies can be trusted if the values (support
  for the president) fall in a normal bell curve
  with most of the people saying they support the
  policies.

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• But what about the situations where the values
  dont fall in a normal bell curve?

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• Then you have untrustworthy results, or at least
  you know that more than you would expect dont
  fit the normal pattern. In the graph at top, most
  of the values fell to the left of center. In
  other words, most of the values are outside the
  normal range.

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Margin of Error

• Margin of Error deserves better than the
  throw-away line it gets in the bottom of stories
  about polling data. Writers who don't understand
  margin of error, and its importance in
  interpreting scientific research, can easily
  embarrass themselves and their news
  organizations.

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Margin of Error

• The margin of error is what statisticians call a
  confidence interval. The math behind it is much
  like the math behind the standard deviation. So
  you can think of the margin of error at the 95
  percent confidence interval as being equal to two
  standard deviations in your polling sample.
  Occasionally you will see surveys with a 99
  percent confidence interval, which would
  correspond to 3 standard deviations and a much
  larger margin of error because the more you
  include the fringe, the more likely your results
  will be untrustworthy.

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Margin of Error

• Lets consider a particular week's poll as a
  repeat of the previous week's. In the first week,
  Candidate A received support from 57 of those
  polled. Candidate B received 43, a 14 point
  difference. In the second week, Candidate A
  received 53 support and Candidate B received
  47, a 6 point difference. Both polls had a
  margin of error of 4 points. So, is Candidate B
  gaining on Candidate A?
• No. Statistically, there is no change from the
  previous week's poll. Politician B has made up no
  measurable ground on Politician A because the
  movement for both politicians is within the 4
  point margin of error.

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Questions Journalists Should Ask

• Where did the data come from? Always ask this one
  first. You always want to know who did the
  research that created the data you're going to
  write about. Just because a report comes from a
  group with a vested interest in its results
  doesn't guarantee the report is a sham. But you
  should always be extra skeptical when looking at
  research generated by people with a political
  agenda. At the least, they have plenty of
  incentive NOT to tell you about data they found
  that contradict their organization's position.

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Questions

• Have the data been peer-reviewed? If it was, you
  know that the data you'll be looking at are at
  least minimally reliable because other pollsters
  have given their blessing on the data. If it
  wasnt, thats a sign that it might not be valid
  data.

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Questions

• How were the data collected? This one is real
  important to ask, especially if the data were not
  peer-reviewed. If the data come from a survey,
  for example, you want to know that the people who
  responded to the survey were selected at random.

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Questions

• Be skeptical when dealing with comparisons.
  Researchers like to do something called a
  "regression," a process that compares one thing
  to another to see if they are statistically
  related. They will call such a relationship a
  "correlation." Always remember that a correlation
  DOES NOT mean causation.

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Questions

• Finally, be aware of numbers taken out of
  context. Again, data that are "cherry picked" to
  look interesting might mean something else
  entirely once it is placed in a different context.

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Survey Sample Sizes

• The population of a study is everyone who could
  have been included. For a national poll, then,
  the population would include every adult in the
  U.S. a number that would be impractical to
  poll. Some researchers take a random sample. The
  larger the sample the more likely it will be
  representative of the population. But a sample of
  400 is usually good enough for most surveys. Most
  national polls, though, survey 1,500 to 2,500
  people. The margin of error in a sample 1
  divided by the square root of the number of
  people in the sample

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Survey Sample Sizes

• The margin of error in a sample 1 divided by
  the square root of the number of people in the
  sample
• In a survey of 2,500 people, the square root is
  50. So, 1/50 .02
• In a survey of 400 people, the square root is 20.
  So, 1/20 .05
• This shows the margin of error increases
  significantly as the number surveyed decreases.

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Picking the Right Statistical Test

• There are different kinds of stats tests and the
  correct one will be the one that provides the
  best answers based on the type of data you have
  collected.
• It is best to enlist the help of a statistics pro
  to analyze your data.
• You can also use SPSS, a computer program that
  conducts the statistical computations for you
  when you enter the data. So by knowing what type
  of test to run, you can enter the data into SPSS
  and run the test.

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Use of Statistics

• Statistical tests allow researchers to find out
  whether their findings are significant i.e.
  What is the probability that what we think is a
  relationship between two variables is really just
  a chance occurrence? The lower the probability of
  chance, the more believable the results.
• Researchers hypothesize. They write a statement
  that they believe will be true from the data they
  collect. They base this on previous research and
  on common sense. Then, they write the null
  hypothesis. The null is the exact opposite of
  the hypothesis the researcher has chosen. The
  statistical tests are done to test whether the
  null hypothesis is correct. If it is WRONG, then
  the researchers hypothesis must be correct.

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Use of Statistics

• Researchers use statistics to determine the
  probability of the data being correct. They
  usually want a confidence level of .05 and it is
  written p .05 That means that the data will
  be 95 percent accurate. (In other words, if the
  data were collected 100 more times, the results
  would fall within the range of the current study
  95 times.) That means the data are pretty
  reliable.

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ANOVA

• Most common statistical test Analysis of
  Variance (ANOVA) is a statistical technique that
  is used to compare the means of more than two
  groups. There are One-way ANOVA (one dependent
  variable and one independent variable) and
  Two-way ANOVA (one dependent and two independent
  variables). Note about variables the
  dependent variable (say, choice of candidate) is
  what will be affected by the question or the
  experiment the independent variables are
  controlled by the researcher (say, choosing
  gender or income as factors that affect the
  dependent variable choice of candidate).
• Use the ANOVA test only if you are comparing data
  from at least 3 groups.

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T-test

• Another common statistical test t-test uses the
  standard deviation of the sample to help
  determine interesting stuff about the larger
  population.
• Use when you have only 2 groups of data, say
  results from men and women and you want to know
  whether their answers are significantly different
  or just from random chance.

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Other types of tests

• There are many other types of tests for
  interpreting data that require a rather high
  level of skill in statistics. If your data are
  complicated and you want to find out as much
  about the data as possible, you may want to
  consult a stats pro for help.