Statistics

In this lesson you will: Match the summary to the shape of the data, and say why you chose it; Pull the right total out of a two-way table for the question actually asked; Separate what a line of fit predicts from what the data can support.

One variable: the shape decides the summary

Ohio limits the displays to dot plots, histograms and box plots, and the reason they keep appearing is that each one answers a different question. A dot plot shows every value. A histogram shows shape. A box plot shows position and spread, and it is the one to reach for when comparing two groups.

Then pick the summary. A roughly symmetric set is fairly described by the mean and the standard deviation. A skewed set, or one with an outlier, is better described by the median and the interquartile range, because neither is moved much by an extreme value.

The 1.5rule flags an outlier: anything below Q_1-1.5or above Q_3+1.5.

Two variables: table, plot, line

A two-way table holds three different kinds of percentage, and the wording of the question tells you which one is wanted. Divide by the grand total for a joint frequency, by a row total or a column total for a conditional one. An association shows up as a gap between two conditional percentages.

For numerical pairs, the scatter plot comes first and the line comes second. Describe direction, form and strength before you fit anything — a line through data with no linear pattern is a correct calculation and a wrong answer.

In y-hat=mx+b, the slope is the predicted change in y for a one-unit increase in x, and b is the prediction at x=0, which is often outside the data.

What the number does not say

A correlation coefficient measures how close the points sit to a straight line, and nothing else. A value near 1 or -1 is a tight linear pattern; a value near 0 means no linear pattern, which is not the same as no pattern at all.

And a strong correlation is not a cause. Two quantities can move together because one drives the other, because both follow a third thing, or by accident in a small sample.

Predicting outside the range of the data is the other quiet failure. The model has no information out there, and it will happily return a negative population or a score above the maximum.

Common mistake — Taking a two-way table percentage from the wrong total

A table shows 48 bus riders late at least once out of 120 bus riders, in a school of 200 students. Asked what percent of bus riders were late, a student writes (48)/(200)=24%.

The grand total is the number printed in the corner, it is the largest number on the table, and it feels like the total. So it gets used.

But the question named its own total. \"Of the bus riders\" means the denominator is 120, and the answer is (48)/(120)=40%. The 24% answers a different question: what percent of all students are bus riders who were late.

How to catch it: Before you divide, say the denominator out loud: \"out of all bus riders\" or \"out of the whole school.\" The phrase \"of the\" in the question names it. If the question has no such phrase, it wants the grand total.