Statistical Analysis and Inference

The correlation coefficient 相関係数 is an index that expresses, as a number from −1 to +1, which way and how strongly two variables move together in a linear relationship.

A positive value means positive correlation, where one variable tends to rise as the other rises, and a negative value means negative correlation, where one tends to fall as the other rises; the sign shows the direction, and the size of the absolute value shows the strength. A scatter plot 散布図 places each pair of values as a dot on a plane with a vertical and a horizontal axis, and the closer the absolute value is to 1, the more closely the dots line up along a straight line.

How to rememberIt is often written with the symbol r.

An explanatory variable 説明変数 is a variable on the cause side (input side), used to explain or predict an outcome.

It is also called an independent variable 独立変数, and in regression analysis it is the value you plug into the equation. If you predict drink sales from the temperature, for example, the temperature is the explanatory variable.

A target variable 目的変数 is a variable on the outcome side (output side) that you want to predict or explain using explanatory variables.

It is also called a dependent variable 従属変数, and in regression analysis it is the value the equation produces. The quantity you actually want to find out is the one you set as the target variable.

AnalogyThink of the treasure chest at the end of a treasure hunt. You follow the map and clues (explanatory variables), and the place you are trying to reach is the target variable.

Estimation 推定 is a method of working out values such as the mean of the whole group you want to study (the population 母集団) from data on a sample 標本, a part taken from that group.

There is point estimation 点推定, which gives a single value, and interval estimation 区間推定, which gives a range with a width. Interval estimation also states how likely it is that the population value falls in that range, as a confidence level 信頼度.

AnalogyIt is like tasting one spoonful of soup to judge how salty the whole pot is. The spoonful is the sample, and the whole pot is the population.

How to rememberThe Japanese word 推定 combines 推 (to infer) and 定 (to fix). You start from the part in hand and pin down a value for the whole.

Hypothesis testing 仮説検定 is a method in which you set up a hypothesis you hope to reject (the null hypothesis 帰無仮説), check how unlikely the sample data would be if it were true, and decide whether to reject it.

The probability used as the threshold is called the significance level 有意水準, and it is set in advance, at 5% or 1% for example. If the data would be less likely than the significance level under the null hypothesis, you reject 棄却 it and adopt the hypothesis you actually want to claim (the alternative hypothesis 対立仮説).

AnalogyIt resembles the courtroom rule of "innocent until proven guilty." You start by assuming innocence (the null hypothesis) and find the defendant guilty only when the evidence is too strong to explain that way.

How to rememberThe Japanese term 帰無 means "to return to nothing." Remember it as a hypothesis set up from the start in the hope of being thrown out.

Regression analysis 回帰分析 is an analysis method that expresses the relationship between a target variable and explanatory variables as a regression equation 回帰式, and uses the explanatory variable values to predict the target variable.

With one explanatory variable it is called simple regression analysis 単回帰分析, and with two or more it is called multiple regression analysis 重回帰分析. Simple regression finds the straight line y = ax + b that best fits the dots on a scatter plot and uses it for prediction.

AnalogyIt is like drawing a single path through scattered footprints, then following that path to guess where the next step will land.

How to rememberThe name "regression" comes from a study of parents' and children's heights, which found that children's heights tend to move back (regress) toward the average.

Correlation analysis 相関分析 is an analysis method that uses the correlation coefficient 相関係数 and a scatter plot 散布図 to examine how closely two variables are related.

You first draw a scatter plot to see how the dots are arranged, then use the correlation coefficient to put the strength of the relationship into a number. The data are collected as pairs of values, such as the height and weight measured from the same person.