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Statistics for Science Journalists

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Statistics for Science Journalists

Steve Doig

Cronkite School of Journalism

- Definition of journalist: A do-gooder who hates math.
- “Word person, not a numbers person.”
- 1936 JQ article noting habitual numerical errors in newspapers
- Japanese 6th graders more accurate on math test than applicants to Columbia’s Graduate School of Journalism
- 20% of journalists got more than half wrong on 25-question “math competency test” (Maier)
- 18% of 5,100 stories examined by Phil Meyer had math errors

- Paulos: 300% decrease in murders
- Detroit Free Press (2006): Compared ACS to Census data to get false drop in median income
- KC Star (2000): Priests dying of AIDS at 4 times the rate of all Americans
- Delaware ZIP Code of infant death
- NYT: 51% of women without spouses

- Numbers that don’t add up
- Making the reader do the math
- Failure to ask “Does this make sense?”
- Over-precision
- Ignoring sampling error margins
- Implying that correlation equals causation

- Misleads math-challenged readers/viewers
- Hurts credibility among math-capable readers/viewers
- Leads to charges of bias, even when cause is ignorance
- Makes reporters vulnerable to being used for the agendas of others

- Randomized experiments: Measure deliberate manipulation of the environment
- Observational studies: Measure the differences that occur naturally
- Meta-analyses: Quantitative review of multiple studies
- Case Study: Descriptive in-depth examination of one or a few individuals

...don’t exist!

- Variable measurements include unpredictable errors or discrepancies that aren’t easily explained.
- Natural variabilityis the result of the fact that individuals and other things are different.

- Measurement error
- Natural variability between individuals
- Natural variability over time in a single individual
Statistics are tools to help us work with measurements that vary

If you found a wallet with $20, would you:

- “Keep it?”
(23% would keep it)

- “Do the honest thing and return it?”
(13% would keep it)

- “Do you use drugs?”
- “Are you religious?”

People routinely say they have voted when they actually haven’t, that they don’t smoke when they do, and that they aren’t prejudiced.

One study six months after an election:

- 96% of actual voters said they voted.
- 40% of non-voters said they voted.

Washington Post poll : “Some people say the 1975 Public Affairs Act should be repealed. Do you agree or disagree that it should be repealed?”

- 24% said yes
- 19% said no
- rest had no opinion

Later Washington Post poll: “President Clinton says the 1975 Public Affairs Act should be repealed. Do you agree or disagree that it should be repealed?”

- 36% of Democrats agreed
- 16% of Republicans agreed
- rest had no opinion

- “Do you support our soldiers in Iraq so that terrorists won’t strike the U.S. again?”

- “About how many times a month do you normally go out on a date?”
- “How happy are you with life in general?”

95% of the time, a random sample’s characteristics will differ from the population’s by no more than about

where N= sample size

- The larger the sample, the smaller the margin of sampling error.
- The size of the population being surveyed doesn’t matter.*
*Unless the sample is a significant fraction of the population.

- Bigger sample means more cost (money and/or time)
- Diminishing return on error margin improvement as sample increases.
- N=100: +/- 10 percentage points
- N=400: +/- 5 percentage points
- N=900: +/- 3.3 percentage points

- Sample needs only to be large enough to give a reasonable answer.
- Sampling error affects subsamples, too.

- The Center
- The Variability
- The Shape

- Mean (average): Total of the values, divided by the number of values
- Median: The middle value of an ordered list of values
- Mode: The most common value
- Outliers: Atypical values far from the center

- Average: $2,827,104
- Median: $950,000
- Mode: $327,000 (also the minimum)
- Outlier: $21.7 million (Alex Rodriguez of the NY Yankees)

Some measures of variability:

- Maximum and minimum: Largest and smallest values
- Range: The distance between the largest and smallest values
- Quartiles: The medians of each half of the ordered list of values
- Standard deviation: Think of it as the average distance of all the values from the mean.

- Don’t consider the average to be “normal”
- Variability is normal
- Anything within about 3 standard deviations of the mean is “normal”

mean

- Symmetrical (not skewed)
- One peak in the middle, at the mean
- The wider the curve, the greater the standard deviation
- Area under the curve is 1 (or 100%)

Your percentilefor a particular measure (like height or IQ) is the percentage of the population that falls belowyou.

Compared to other American males:

- My height (5’ 11”): 75th percentile
- My weight (230 lbs.): 85thpercentile
- My age (64): 87th percentile
Therefore, I am older and heavier than I am tall.

A standardized score(also called the z-score) is simply the number of standard deviations a particular value is either above or below the mean.

The standardized score is:

- Positive if above the mean
- Negative if below the mean
Useful for defining data points as outliers.

For any normal curve, approximately:

- 68% of values within one StdDevof the mean
- 95% of values within two StdDevsof the mean
- 99.7% of values within three StdDevsof the mean

- A value that is more than three standard deviations above or below the mean.

Correlation (also called the correlation coefficient or Pearson’s r) is the measure of strength of the linear relationship between two variables.

Think of strength as how closely the data points come to falling on a line drawn through the data.

- Correlation can range from +1 to -1
- Positive correlation: As one variable increases, the other increases
- Negative correlation: As one variable increases, the other decreases
- Zero correlation means the best line through the data is horizontal
- Correlation isn’t affected by the units of measurement

r = +.4

r = +1

r = +.8

r = +.1

r = -.4

r = -.1

r = -.8

r = -1

r = 0

r = 0

r = .8

r = .8

Correlation does not imply causation.

(Churches and liquor stores, shoe size and reading ability)

- When considering relationships between measurement variables, there are two kinds:
- Explanatory (or independent) variable: The variable that attempts to explain or is purported to cause (at least partially) differences in the…
- Response (or dependent or outcome) variable

- Often, chronology is a guide to distinguishing them (examples: baldness and heart attacks, poverty and test scores)

- The explanatory variable is the direct cause of the response variable
Example: pollen counts and percent of population suffering allergies, intercourse and babies

- The response variable is causing a change in the explanatory variable
Example: hotel occupancy and advertising spending, divorce and alcohol abuse

- The explanatory variable is a contributing -- but not sole -- cause
Example: birth complications and violence, gun in home and homicide, hours studied and grade, diet and cancer

- Both variables may result from a common cause
Example: SAT score and GPA, hot chocolate and tissues, storks and babies, fire losses and firefighters, WWII fighter opposition and bombing accuracy

- Both variables are changing over time
Example: divorces and drug offenses, divorces and suicides

- The association may be nothing more than coincidence
Example: clusters of disease, brain cancer from cell phones

The only way to confirm is with a designed (randomized double-blind) experiment.But non-statistical evidence of a possible connection may include:

- A reasonable explanation of cause and effect.
- A connection that happens under varying conditions.
- Potential confounding variables ruled out.

In addition to figuring the strength of the relationship, we can create a simple equation that describes the best-fit line (also called the “least-squares” line) through the data.

This equation will help us predict one variable, given the other.

x = horizontal axis y = vertical axis

Equation for a line:

y = slope * x + intercept

or as it often is stated:

y =mx + b

- Public school test scores
- Cheating in school test scores
- Tenure of white vs. black coaches in NBA
- Racial bias in picking jurors
- Racial profiling in traffic stops

Confusing these two:

- Probability of actually having a condition, given a positive test for it
- Probability of having a positive test, given actually having the condition
When the incidence of some disease or condition is very low, and the test for it is not perfect, there will be a high probability that a positive test result is false positive.

- Base rate: The probability that someone has a disease or condition, without knowing any test results.
- Test Sensitivity: Proportion of people who correctly test positive when they have the disease or condition (true positive)
- Test Specificity: Proportion of people who correctly test negativewhen they don’t have the disease or condition (true negative)

Consider this scenario:

- Base rate: 1% of population to be tested uses dangerous drugs
- You use a test that’s 99% accurate in both sensitivity and specificity
- 10,000 people are tested

TestTestTotal

PositiveNegative

Users 100

Not 9,900

Total 10,000

TestTestTotal

PositiveNegative

Users 991100

Not 9,900

Total 10,000

TestTestTotal

PositiveNegative

Users991100

Not9,8019,900

Total9,80210,000

TestTestTotal

PositiveNegative

Users991100

Not???9,8019,900

Total9,80210,000

TestTestTotal

PositiveNegative

Users991100

Not 9,8019,900

Total1989,80210,000

(50% of positives are FALSE!)

99

- Like the error margin around poll results
- A confidence interval is a tradeoff between certainty and accuracy, like shooting at targets of different sizes
- The bigger the sample, the smaller the confidence interval at the 95% level
- When comparing results, if confidence intervals overlap, the results are NOT statistically significant

- P-value is the probability that the sample result is significantly different from the true result (i.e., wrong)
- 95% confidence interval (p < 0.05) is the most commonly used interval in social science research
- Hard science, particularly medicine, often needs tighter confidence intervals and smaller p-values, like p<0.01
- Studies are going to be wrong about 5% of the time (and you won’t know when)
- On the other hand, they probably won’t be very wrong.

- Pay attention to the method: Observational, randomized double-blind experiment, meta-analysis, case study
- Note the sample size
- Don’t ignore the confidence intervals
- Consider the p-value as the probability you’re writing about something that isn’t true
- Remember correlation doesn’t necessarily mean causation.
- Consider the quality of the journal (peer reviewed?)
- Who paid for the research?

- “Numbers in the Newsroom”, by Sarah Cohen, IRE
- “News and Numbers”, by Victor Cohn and Lewis Cope
- “Precision Journalism (4th edition)”, by Phil Meyer
- “Innumeracy”, by John Allen Paulos
- “A Mathematician Reads the Newspaper,” by John Allen Paulos