Quotes from Charles Wheelan

Here is a statement of the obvious: banks with "state" in the name are chartered by the state; those with "national" in their name are chartered by the feds.)

The whole point of a representative sample is that it looks like the underlying population.

Specifically, the sample means will form a normal distribution around the population mean, which in this case is $70,900.

The larger the number of samples, the more closely the distribution will approximate the normal distribution.

If there are 60,000 blue marbles and 40,000 red marbles in a giant urn, then the most likely composition of a sample of 100 marbles drawn randomly from the urn would be 60 blue marbles and 40 red marbles.

The world is producing more and more data, ever faster and faster. Yet, as the New York Times has noted, "Data is merely the raw material of knowledge."3* Statistics is the most powerful tool we have for using information to some meaningful end

some might have 62 blue marbles and 38 red marbles, or 58 blue and 42 red. But the chances of drawing any random sample that deviates hugely from the composition of marbles in the urn are very, very low.

we have now introduced two different measures of dispersion: the standard deviation and the standard error.

The standard deviation measures dispersion in the underlying population

The standard error measures the dispersion of the sample means.

these cases, the goal is to find two groups of subjects who are broadly similar except for the application of whatever "treatment" we care about.

Life gets a little trickier when we are doing our regression analysis (or other forms of statistical inference) with a small sample of data.

Logic suggests that we should be less confident about generalizing our results to the entire adult population from a sample of 25 than from a sample of 3,000.

Our sample of 25 will still give us meaningful information, as would a sample of 5 or 10—but how meaningful? The t-distribution answers that question.

They will still be distributed around the true coefficient for the whole population, but the shape of that distribution will not be our familiar bell-shaped normal curve.

Several of the high schools consistently at the top of the rankings are selective enrollment schools, meaning that students must apply to get in, and only a small proportion of those students are accepted.

Instead, we have to assume that repeated samples of just 25 will produce more dispersion around the true population coefficient—and therefore a distribution with "fatter tails.

So let's summarize: (1) these schools are being recognized as "excellent" for having students with high test scores; (2) to get into such a school, one must have high test scores. This is the logical equivalent of giving an award to the basketball team for doing such an excellent job of producing tall students.

The t-distribution is actually a series, or "family," of probability density functions that vary according to the size of our sample.

A statistical anomaly does not prove wrongdoing. Delma Kinney, a fifty-year-old Atlanta man, won $1 million in an instant lottery in 2008 and then another $1 million in an instant game in 2011. The probability of that happening to the same person is somewhere in the range of 1 in 25 trillion.

You go to war with the army you have—not the army you might want or wish to have at a later time.

Specifically, the more data we have in our sample, the more "degrees of freedom" we have when determining the appropriate distribution against which to evaluate our results.

Statistics cannot prove anything with certainty. Instead, the power of statistical inference derives from observing some pattern or outcome and then using probability to determine the most likely explanation for that outcome.

For large samples, we can assume that the standard deviation of the sample is reasonably close to the standard deviation of the population.*