Chapter 1-1. Quantile-Quantile Plot (Q-Q plot)

Higher category: 【Statistics】 Chapter 1. Basics of Statistics

1. Overview

⑴ Maximum, Minimum, Range

① Maximum: the largest value

② Minimum: the smallest value

③ Range = Maximum - Minimum

④ Midrange = (Maximum + Minimum) / 2

⑵ Quantile

① Quantile function: the inverse of the cumulative distribution function Φ

○ Domain: {x 0 ≤ x ≤ 1}

○ Range: the statistic of the population of interest

② Depending on the number of intervals, there are percentiles (100 quantiles), quartiles (4 quantiles), etc.

③ Median: 50th percentile

④ First quartile: 25th percentile, that is, the 25th smallest number out of 100

⑤ Third quartile: 75th percentile, that is, the 75th smallest number out of 100 (or the 25th largest number)

Figure. 1. quantile-quantile plot (Q-Q plot)

⑴ Definition: a set of points (x, y) as follows

① ｛(x, y) | Φ = P(X ＜ x) = P(Y ＜ y)｝

② note that x is the statistic of the standard normal distribution and y is the statistic of the sample group

⑵ Case study

① when the sample group follows a normal distribution: Q-Q plot is close to a straight line

② if the distribution of samples is skewed to the right

○ Skewness ＜ 0

○ idea: you can think of dragging each point to the right from the existing normal distribution

○ since the random variable of each point increases, each point on the Q-Q plot moves upward from the diagonal

③ if the distribution of samples is skewed to the left

○ Skewness ＞ 0

○ idea: you can think of dragging each point to the left from the existing normal distribution

○ since the random variable of each point decreases, each point on the Q-Q plot moves downward from the diagonal

Input: 2019.10.10 11:50