Describing data: center, spread and the normal curve

Transcript

RyanHere's a little puzzle from the OpenStax statistics textbook. You have fifty incomes. Forty-nine of them are thirty thousand dollars. One is five million dollars. What's the typical income?

SoheeIf you add them all up and divide, the average comes out far above thirty thousand. And yet forty-nine of the fifty people earn exactly thirty thousand.

RyanSo which number tells the truth? That's today.

SoheeThis is EPPP Prep, the podcast. I'm Sohee.

RyanAnd I'm Ryan. Today we're describing data: the center, the spread, the shape, and the normal curve. And we promise, no reading formulas aloud. We'll stick to what each number means.

SoheeStart with the two branches of statistics. The statistics text says organizing and summarizing data is called descriptive statistics. Drawing conclusions from data, using probability, is inferential statistics.

RyanAnd two words that go with them. A statistic is a number describing a sample. A parameter is a numerical characteristic of the whole population, which a statistic estimates.

SoheeBefore you can describe anything, you need to know how it was measured. The text names four levels of measurement, from lowest to highest.

RyanNominal is categories with no order. Favorite food, or yes or no. Ordinal has an order, but you can't measure the differences. Think of ranking the top five national parks, or ratings like excellent, good and satisfactory.

SoheeInterval data are numbers where differences make sense, but there's no true zero. The textbook's example is temperature in Celsius or Fahrenheit. Zero degrees isn't the lowest possible temperature. It can get colder than zero.

RyanAnd ratio data have a true minimum, a zero, so ratios can be calculated. The textbook uses exam scores out of a hundred points.

SoheeWhy does this matter? The text says correct statistical procedures depend on it. Not every statistical operation can be used with every set of data.

RyanNow to the center. Three measures. The psychology text defines them neatly. The mode is the most frequent response, the median lies at the middle of the data set, and the mean is the arithmetic average.

SoheeAnd it adds a warning. Means are very sensitive to the effects of outliers.

RyanWhich solves our puzzle. The statistics text says the median is a better measure of center there, because forty-nine values are thirty thousand and one is five million. The five million is an outlier. The thirty thousand gives a better sense of the middle.

SoheeThe text's general rule: the median is generally the better measure of center when there are extreme values or outliers, because it isn't affected by their precise values.

RyanAnd the mode has a special talent. It works for qualitative data. If most people in a set chose red, the mode is red. You can't take the mean of colors.

SoheeNow shape. In a perfectly symmetrical distribution, the mean and median are the same. But when the data have a long tail, things separate.

RyanThe text says when a distribution is skewed to the right, the mean is often greater than the median. Skewed to the left, the mean is often less than the median.

SoheeAnd here's a picture to hold onto. The text notes the median tends to stay closest to the high point, the mode, while the mean tends to sit farther out on the tail.

RyanSo the mean gets dragged toward the tail. In a right skew, also called a positive skew, a few very high values pull the mean up above the median.

SoheeOkay, pause and think. In a set of scores, the mean is noticeably higher than the median. Which way is the distribution most likely skewed?

RyanTo the right, the positive direction. The mean has been pulled out toward a tail of high values.

SoheeNow spread. The text calls the standard deviation the most common measure of spread, a number that measures how far data values are from their mean.

RyanSmall when data cluster close to the mean. Large when they're spread out. And it's never negative. A standard deviation of zero means every value is identical.

SoheeIts partner is the variance. The intuition: take each value's distance from the mean, square those distances so they're all positive, and average them. That's the variance.

RyanWhy square them? The text explains that if you add up the raw deviations, the sum is always zero. Squaring fixes that.

SoheeBut squaring changes the units. So the standard deviation is the square root of the variance, which brings the spread back into the same units as the data.

RyanOne more spread measure. The interquartile range, which the text describes as the spread of the middle half of the data. It's handy for spotting potential outliers.

SoheeNow the bridge to the normal curve. The z-score. The text says a z-score tells you how many standard deviations a value is above or below the mean.

RyanPositive z, above the mean. Negative z, below. A z of zero means exactly at the mean. If the mean is five and the standard deviation is two, a value of eleven is three standard deviations above.

SoheeAnd the beauty of z-scores, in the text's words, is that they let us compare data that are scaled differently. Two people on different tests can each be two of their own standard deviations above their own mean.

RyanIf the data are normally distributed, the empirical rule tells you roughly where everything sits. About 68 percent of values fall within one standard deviation of the mean.

SoheeAbout 95 percent within two. And about 99.7 percent within three. Almost everything.

RyanNext, percentiles, and this one matters for all of us as test takers. The text says scoring at the 90th percentile does not necessarily mean you got 90 percent correct.

SoheeIt means 90 percent of scores are the same as or less than yours. A percentile is a position compared with other people, not a percent of items correct.

RyanPause and think, second one. On a normally distributed test, a client's score has a z of positive two. Roughly how unusual is that?

SoheeAbout 95 percent of scores fall within two standard deviations of the mean, on either side. So a score two above the mean is higher than most. Only a small share of people score beyond it in that direction.

RyanLet's finish with probability, because the knowledge statement names probability theory. The text defines the probability of an outcome as its long-term relative frequency.

SoheeProbabilities run from zero, never happens, to one, always happens. Point five means equally likely to happen or not.

RyanAnd the law of large numbers. As you repeat an experiment more and more times, the observed relative frequency gets closer and closer to the theoretical probability. Flip a fair coin a handful of times and you might get all heads. Flip it thousands of times, and heads gets close to half.

SoheeWhich leads to one more big idea, the central limit theorem. If you repeatedly draw samples of a given size and calculate their means, those means tend to follow a normal distribution, and more so as the samples get bigger.

RyanAnd the spread of those sample means is smaller than the spread of the original scores. The text calls it the standard error of the mean, describing how far, on average, a sample mean will be from the population mean.

SoheeThat's why averages from bigger samples are steadier than single scores. It's the foundation for everything in inferential statistics.

RyanSo how does the exam test this? Expect questions like: which measure of center is best for skewed data, what does a z-score or percentile mean, or what share of a normal distribution falls within certain limits.

SoheeThe tempting wrong answers: choosing the mean for heavily skewed data, reading a percentile as percent correct, and forgetting that interval scales have no true zero.

RyanRecap. Descriptive statistics summarize. Inferential statistics draw conclusions using probability.

SoheeNominal, ordinal, interval, ratio, in that order. Interval has equal differences but no true zero. Ratio has a true zero.

RyanThe mean is pulled toward outliers and the tail. The median resists them. The mode works even for categories.

SoheeStandard deviation is spread in the original units, and variance is its square. A z-score counts standard deviations from the mean. And the normal curve gives roughly 68, 95 and 99.7 percent within one, two and three standard deviations.

RyanProbability is long-run relative frequency, and sample means settle into a normal shape as samples grow.

SoheeIf numbers make your stomach tighten, you are in good company. Today wasn't about calculation. It was about meaning, and meaning is what these questions test.

RyanTry a few descriptive statistics questions next, and sketch a quick curve on your scratch paper as you go. Thanks for listening to EPPP Prep.

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