Quantile-Quantile (q-q) Plots

Author(s)

David

Scott

Prerequisites

Histograms,Distributions,Percentiles,Describing

Bivariate Data,Normal

Distributions

Introduction

The quantile-quantile or q-q plot is an exploratory graphical

device used to check the validity of a distributional assumption

for a data set. In general, the basic idea is to compute the

theoretically expected value for each data point based on the

distribution in question. If the data indeed follow the assumed

distribution, then the points on the q-q plot will fall

approximately on a straight line.

Before delving into the details of q-q plots, we first describe two

related graphical methods for assessing distributional assumptions:

the histogram and the cumulative distribution function

(CDF). As will be seen, q-q plots are more general than

these alternatives.

Assessing Distributional Assumptions

As an example, consider data measured from a physical device such

as the spinner depicted in Figure 1. The red arrow is spun around

the center, and when the arrow stops spinning, the number between 0

and 1 is recorded. Can we determine if the spinner is fair?

Figure 1. A physical device that gives samples from a uniform

distribution.

If the spinner is fair, then these numbers should follow a uniform

distribution. To investigate whether the spinner is fair, spin the

arrow n times, and record the measurements by {μ1,

μ2, ..., μn}. In this example, we collect n =

100 samples. The histogram provides a useful visualization of these

data. In Figure 2, we display three different histograms on a

probability scale. The histogram should be flat for a uniform

sample, but the visual perception varies depending on whether the

histogram has 10, 5, or 3 bins. The last histogram looks flat,

but the other two histograms are not obviously flat. It is not

clear which histogram we should base our conclusion on.

Figure 2. Three histograms of a sample of 100 uniform points.

Alternatively, we might use the cumulative distribution function

(CDF), which is denoted by F(μ). The CDF gives the probability that

the spinner gives a value less than or equal to μ, that is, the

probability that the red arrow lands in the interval [0, μ]. By

simple arithmetic, F(μ) = μ, which is the diagonal straight line y

= x. The CDF based upon the sample data is called the empirical

CDF (ECDF), is denoted by , and is

defined to be the fraction of the data less than or equal to μ;

that is, 

In general, the ECDF takes on a ragged staircase

appearance. For the spinner sample analyzed in Figure 2, we computed the ECDF

and CDF, which are displayed in Figure 3. In the left frame, the

ECDF appears close to the line y = x, shown in the middle frame. In

the right frame, we overlay these two curves and verify that they

are indeed quite close to each other. Observe that we do not need

to specify the number of bins as with the histogram.

Figure 3. The empirical and theoretical cumulative distribution

functions of a sample of 100 uniform points.

q-q plot for uniform data

The q-q plot for uniform data is very similar to the empirical CDF

graphic, except with the axes reversed. The q-q plot provides a visual comparison of the sample

quantiles to the corresponding theoretical quantiles. In

general, if the points in a q-q plot depart from a straight line,

then the assumed distribution is called into question.

Here we define the qth quantile of a batch of n numbers as a number

ξqsuch that a fraction q x n of the sample is less than

ξq, while a fraction (1 - q) x n of the sample is

greater than ξq. The best known quantile is the median,

ξ0.5, which is located in the middle of the sample.

Consider a small sample of 5 numbers from the

spinner: μ1 =

0.41, μ2 =0.24,

μ3 =0.59,

μ4 =0.03,and

μ5 =0.67.

Based upon our description of the spinner, we expect a uniform

distribution to model these data. If the sample data were

“perfect,” then on average there would be an observation in the

middle of each of the 5 intervals: 0 to .2, .2 to .4, .4 to .6, and

so on. Table 1 shows the 5 data points (sorted in ascending order)

and the theoretically expected value of each based on the

assumption that the distribution is uniform (the middle of the

interval).

Table 1. Computing the Expected Quantile Values.

Data (μ)

Rank (i)

Middle of the ith Interval

.03

.24

.41

.59

.67

1

2

3

4

5

.1

.3

.5

.7

.9

The theoretical and empirical CDFs are shown in Figure 4 and the

q-q plot is shown in the left frame of Figure

5.

Figure 4. The theoretical and empirical CDFs of a small sample of 5

uniform points, together with the expected values of the 5 points

(red dots in the right frame).

In general, we consider the full set of sample quantiles to be the

sorted data values

μ(1) <

μ(2) <

μ(3) < ··· <

μ(n-1) <

μ(n) ,

where the parentheses in the subscript indicate the data have been

ordered. Roughly speaking, we expect the first ordered value to be

in the middle of the interval (0, 1/n), the second to be in the

middle of the interval (1/n, 2/n), and the last to be in the middle

of the interval ((n - 1)/n, 1). Thus, we take as the theoretical

quantile the value

where q corresponds to the ith ordered sample value. We

subtract the quantity 0.5 so that we are exactly in the middle of

the interval ((i - 1)/n, i/n). These ideas are depicted

in the right frame of Figure 4 for our small sample of size n =

5.

We are now prepared to define the q-q plot precisely. First, we

compute the n expected values of the data, which we pair with the n

data points sorted in ascending order. For the uniform density,

the q-q plot is composed of the n ordered pairs

This definition is slightly different from the ECDF, which includes

the points (u(i), i/n). In the left frame of Figure 5,

we display the q-q plot of the 5 points in Table 1. In the right

two frames of Figure 5, we display the q-q plot of the same batch

of numbers used in Figure 2. In the final frame, we add the

diagonal line y = x as a point of reference.

Figure 5. (Left) q-q plot of the 5 uniform points. (Right) q-q plot

of a sample of 100 uniform points.

The sample size should be taken into account when judging how close

the q-q plot is to the straight line. We show two other uniform

samples of size n = 10 and n = 1000 in Figure 6. Observe that the

q-q plot when n = 1000 is almost identical to the line y = x, while

such is not the case when the sample size is only n = 10.

Figure 6. q-q plots of a sample of 10 and 1000 uniform points.

In Figure 7, we show the q-q plots of two random samples that are

not uniform. In both examples, the sample quantiles match the

theoretical quantiles only at the median and at the extremes.

Both samples seem to be symmetric around

the median. But the data in the left frame are closer to the median

than would be expected if the data were uniform. The data in the

right frame are further from the median than would be expected if

the data were uniform.

Figure 7. q-q plots of two samples of size 1000 that are not

uniform.

In fact, the data were generated in the R language from beta

distributions with parameters a = b = 3 on the left and a = b =0.4

on the right. In Figure 8 we display histograms of these two data

sets, which serve to clarify the true shapes of the densities.

These are clearly non-uniform.

Figure 8. Histograms of the two non-uniform data sets.

q-q

plot for normal data

The definition of the q-q plot may be extended to any continuous

density. The q-q plot will be close to a straight line if the

assumed density is correct. Because the cumulative distribution

function of the uniform density was a straight line, the q-q plot

was very easy to construct. For data that is not uniform, the

theoretical quantiles must be computed in a different manner.

Let {z1, z2, ..., zn} denote a

random sample from a normal distribution with mean μ = 0 and standard deviation σ = 1. Let the ordered

values be denoted by

z{1) <

z(2) <

z(3) < ... <

z(n-1)(n).

These n ordered values will play the role of the sample

quantiles.

Let us consider a sample of 5 values from a distribution to see how

they compare with what would be expected for a normal distribution.

The 5 values in ascending order are shown in the first column of

Table 2.

Table 2. Computing the expected quantile values for normal

data.

Data (z)

Rank (i)

Middle of theith Interval

Normal(z)

-1.96

-.78

.31

1.15

1.62

1

2

3

4

5

.1

.3

.5

.7

.9

-1.28

-0.52

0.00

0.52

1.28

Just as in the case of the uniform distribution, we have 5

intervals. However, with a normal distribution the theoretical

quantile is not the middle of the interval but rather the inverse

of the normal distribution for the middle of the interval. Taking

the first interval as an example, we want to know the z value such

that 0.1 of the area in the normal distribution is below z. This

can be computed using the Inverse Normal Calculator as shown in

Figure 9. Simply set the “Shaded Area” field to the middle of the

interval (0.1) and click on the “Below” button. The result is

-1.28. Therefore, 10% of the distribution is below a z value of

-1.28.

Figure 9. Example of the Inverse Normal Calculator for finding a

value of the expected quantile from a normal distribution.

The q-q plot for the data in Table 2 is shown in the left frame of

Figure 11.

In general, what should we take as the corresponding theoretical

quantiles? Let the cumulative distribution function of the normal

density be denoted by Φ(z). In the previous example, Φ(-1.28) =

0.10 and Φ(0.00) = 0.50. Using the quantile notation, if

ξq is the qth quantile of a normal

distribution, then

Φ(ξq)= q.

That is, the probability a normal sample is less

than ξq is

in fact just q.

Consider the first ordered value, z(1). What might we

expect the value of Φ(z(1)) to be? Intuitively, we

expect this probability to take on a value in the interval (0,

1/n). Likewise, we expect Φ(z(2)) to take on a value in

the interval (1/n, 2/n). Continuing, we expect Φ(z(n))

to fall in the interval ((n - 1)/n, 1). Thus, the theoretical

quantile we desire is defined by the inverse (not reciprocal) of

the normal CDF. In particular, the theoretical quantile

corresponding to the empirical quantile

z(i) should be

 for i = 1, 2, ..., n.

The empirical CDF and theoretical quantile construction for the

small sample given in Table 2 are displayed in Figure 10. For the

larger sample of size 100, the first few expected quantiles are

-2.576, -2.170, and -1.960.

Figure 10. The empirical CDF of a small sample of 5 normal points,

together with the expected values of the 5 points (red dots in the

right frame).

In the left frame of Figure 11, we display the q-q plot of the

small normal sample given in Table 2. The remaining frames in

Figure 11 display the q-q plots of normal random samples of size n

= 100 and n = 1000. As the sample size increases, the points in the

q-q plots lie closer to the line y = x.

Figure 11. q-q plots of normal data.

As before, a normal q-q plot can indicate departures from

normality. The two most common examples are skewed data and data

with heavy tails (large kurtosis). In Figure 12, we show normal q-q

plots for a chi-squared (skewed) data set and a Student’s-t

(kurtotic) data set, both of size n = 1000. The data were first

standardized. The red line is again y = x. Notice, in particular,

that the data from the t distribution follow the normal curve

fairly closely until the last dozen or so points on each

extreme.

Figure 12. q-q plots for standardized non-normal data (n =

1000).

q-q plots for normal data with general mean and scale

Our previous discussion of q-q plots for normal data all assumed

that our data were standardized. One approach to constructing q-q

plots is to first standardize the data and then proceed as

described previously. An alternative is to construct the plot

directly from raw data.

In this section, we present a general approach for data that are

not standardized. Why did we standardize the data in Figure 12? The

q-q plot is comprised of the n points

If the original data {zi} are normal, but have an

arbitrary mean μ and standard deviation σ, then the line y = x will

not match the expected theoretical quantiles. Clearly, the linear

transformation

μ + σ ξq

would provide the qth theoretical quantile on the transformed

scale. In practice, with a new data set

{x1,x2,...,xn} ,

the normal q-q plot would consist of the n points

Instead of plotting the line y = x as a reference line, the

line

y = M + s · x

should be composed, where M and s are the sample moments (mean and

standard deviation) corresponding to the theoretical moments μ and

σ. Alternatively, if the data are standardized, then the line y = x

would be appropriate, since now the sample mean would be 0 and the

sample standard deviation would be 1.

Example: SAT Case Study

The SAT case study followed the academic achievements of 105

college students majoring in computer science. The first variable

is their verbal SAT score and the second is their grade point

average (GPA) at the university level. Before we compute

inferential statistics using these variables, we should check if

their distributions are normal. In Figure 13, we display the q-q

plots of the verbal SAT and university GPA variables.

Figure 13. q-q plots for the student data (n = 105).

The verbal SAT seems to follow a normal distribution reasonably

well, except in the extreme tails. However, the university GPA

variable is highly non-normal. Compare the GPA q-q plot to the

simulation in the right frame of Figure 7. These figures are very

similar, except for the region where x ≈ -1. To follow these ideas,

we computed histograms of the variables and their scatter diagram

in Figure 14. These figures tell quite a different story. The

university GPA is bimodal, with about 20% of the students falling

into a separate cluster with a grade of C. The scatter diagram is

quite unusual. While the students in this cluster all have below

average verbal SAT scores, there are as many students with low SAT

scores whose GPAs were quite respectable. We might speculate as to

the cause(s): different distractions, different study habits, but

it would only be speculation. But observe that the raw correlation

between verbal SAT and GPA is a rather high 0.65, but when we

exclude the cluster, the correlation for the remaining 86 students

falls a little to 0.59.

Figure 14. Histograms and scatter diagram of the verbal SAT and GPA

variables for the 105 students.

Discussion

Parametric modeling usually involves making assumptions about the

shape of data, or the shape of residuals from a regression fit.

Verifying such assumptions can take many forms, but an exploration

of the shape using histograms and q-q plots is very effective. The

q-q plot does not have any design parameters such as the number of

bins for a histogram.

In an advanced treatment, the q-q plot can be used to formally test

the null hypothesis that the data are normal. This is done by

computing the correlation coefficient of the n points in the q-q

plot. Depending upon n, the null hypothesis is rejected if the

correlation coefficient is less than a threshold. The threshold is

already quite close to 0.95 for modest sample sizes.

We have seen that the q-q plot for uniform data is very closely

related to the empirical cumulative distribution function. For

general density functions, the so-called probability integral

transform takes a random variable X and maps it to the interval (0,

1) through the CDF of X itself, that is,

Y = FX(X)

which has been shown to be a uniform density. This explains why the

q-q plot on standardized data is always close to the line y = x

when the model is correct. Finally, scientists have used special graph paper for years to make

relationships linear (straight lines). The most common example used

to be semi-log paper, on which points following the formula y =

aebx appear linear. This

follows of course since log(y) = log(a) + bx, which is the equation

for a straight line. The q-q plots may be thought of as being

“probability graph paper” that makes a plot of the ordered data

values into a straight line. Every density has its own special

probability graph paper.