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Journalists hate math

- 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

Bad examples abound

- 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

Common problems

- 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

Dangers of journalistic innumeracy

- 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

Common Research Methods

- 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

Simple Measures...

...don’t exist!

Measurement Variability

- 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.

Reasons for variable measures

- 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

Deliberate Bias?

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)

Unintentional Bias?

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

Desire to Please?

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.

Asking the uninformed?

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

Asking the uninformed?

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

Unnecessary Complexity?

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

Question Order

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

Margin of Error

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

where N= sample size

Two Important Concepts about Error Margin

- 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.

Sampling realities

- 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.

Three Useful Featuresof a Set of Data

- The Center
- The Variability
- The Shape

The Center

- 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

Example: Baseball Salaries

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

The Variability

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.

What is “normal”?

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

Some Characteristics of a Normal Distribution

- 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%)

Percentiles

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.

Standardized Scores

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.

The Empirical Rule

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

Outlier

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

Strength of Relationship

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.

Features of Correlation

- 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

Important!

Correlation does not imply causation.

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

Correlation of variables

- 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)

Some reasons why two variables could be related

- The explanatory variable is the direct cause of the response variable

Example: pollen counts and percent of population suffering allergies, intercourse and babies

Some reasons two variables could be related

- The response variable is causing a change in the explanatory variable

Example: hotel occupancy and advertising spending, divorce and alcohol abuse

Some reasons two variables could be related

- 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

Some reasons two variables could be related

- 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

Some reasons two variables could be related

- Both variables are changing over time

Example: divorces and drug offenses, divorces and suicides

Some reasons two variables could be related

- The association may be nothing more than coincidence

Example: clusters of disease, brain cancer from cell phones

So how can we confirm causation?

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.

Linear Regression

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.

Remember 9th Grade Algebra?

x = horizontal axis y = vertical axis

Equation for a line:

y = slope * x + intercept

or as it often is stated:

y =mx + b

Regression in data journalism

- 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

Confusion of the Inverse

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.

Definitions

- 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)

Drug Tests

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

Drug Tests

Test Test Total

PositiveNegative

Users 99 1 100

Not 9,801 9,900

Total 198 9,802 10,000

(50% of positives are FALSE!)

99

Confidence Intervals

- 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-values

- 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.

How to read a research study

- 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?

Newsroom math bibliography

- “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

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