Empirical Procedures: Why we give the Q-Q Plot a Thumbs Up

However the Histogram is not useful for evaluating the fit of the chosen distribution. When there are a small number of data points, a histogram can be rather ragged. Further, our understanding of the fit depends on the widths of the histogram intervals. But even if the intervals are well chosen, grouping data into cells makes it difficult to compare a histogram to a continuous probability density function.

The Q-Q Plots or Quantile-Quantile Plots overcomes all the limitations of the Histogram plot. Let’s take an example:

Consider a sample data from a population. The construction of Q-Q plots starts with ordering the sample data from smallest to largest. Let k denote the ranking or order number. Therefore k=1 for the smallest and k = n for the largest. The q-q plot is based on the fact that the ordered value of k is an estimate of (k-1/2)/n quantile of the sample data. In other words, the ordered values are close to inverse of cumulative distribution of (k-0.5)/n, where n is the sample size. If the cumulative distribution function belongs to an appropriate known distribution, then the plot of ordered values and the known cumulative distributional values will approximately form a straight line. On the other hand, if the assumed distribution is inappropriate, the points will deviate from the straight line in a systematic manner. Therefore, if we assume that the cdf is from a normal distribution, then obtaining a straight line after the plot confirms that the sample data indeed belongs to the normal population.

The following is the R command for attaining a Quantile-Quantile plot for the variable income which is a part of state dataset.

qqnorm(state$Income, main=”Normal Q-Q Plot”,xlab = “Theoretical Quantiles”, ylab = “Income Quantiles”)

Thus, from the graph we conclude that the variable Income follows a Normal distribution. In Q-Q plot we usually focus on the central part of the linearity rather than the extreme plots. This is because the normal curve is asymptotic to X-axis and so reflects in the extreme points of the straight line plots. In similar manner one could get Quantile-Quantile plots for different distributions using qq Plot functions in R software.

In conclusion, through empirical procedures like Histogram & Q-Q plots, we can identify the general form of the distribution. Among these two, Q-Q plots perform well even for small sample size. As most of our statistical methods depend on the normality assumptions, checking normality for the sample data is important. If we violate this assumption then the inference driven from the analysis may not be precise.

So, though both empirical procedures, the Histogram & Q-Q plots, can identify the general form of the distribution, we can see that the Q-Q plots perform well even for small sample sizes. What is important to remember is that as most of our statistical methods depend on the normality assumptions, checking normality for the sample data is important. If we violate this assumption then the inference driven from the analysis may not be precise.

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