Simple summary statistics

Useful MATLAB commands

There are many inbuilt MATLAB commands for calculating statistical properties of data. Some of them are listed below:

  • max - returns the maximum value of an array.
  • min – returns the minimum value of an array.
  • mean – returns the mean (average) value of the array.
  • median – returns the median value of an array.
  • mode – returns the mode value of an array.
  • std – returns the standard deviation of an array.
  • var – returns the variance of an array.
  • hist – plots the values in the array as a histogram.
  • sort – sorts the elements of the array in ascending order.
  • sum – returns the sum of all the elements of the array.
  • rand – generates uniformly distributed random numbers.
  • randn – generates normally distributed random numbers with mean 0 and variance 1.

The following activities will help you use these basic commands.

Question

Look at the MATLAB help files (doc max etc) for each of the above commands and apply them all (except rand and randn) to the following vector:

x = [3 -47 -6 -29 37 -22 -7 12 -7 -18 -53 -12 34 12 -35 28 -5 22 4 -70 -32 47 -6 -19 8 17 -22 9 -39 -21];

What happens if you apply these to a matrix? For example, try applying them to the following matrix:

randn(100,5)
Expand for solution
Solution

The operations yield the following:

  • max(x) : 47
  • min(x) : -70
  • mean(x) : -7.2333
  • median(x) : -6.5
  • mode(x) : -22
  • std(x) : 27.6314
  • var(x) : 763.4954
  • hist(x) : Histogram example
  • sort(x) : 
    >> sort(x)
ans =
Columns 1 through 17
-70   -53   -47   -39   -35   -32   -29   -22   -22   -21   -19   -18   -12    -7    -7    -6    -6
Columns 18 through 30
    -5     3     4     8     9    12    12    17    22    28    34    37    47
  • sum(x) : -217

Applying them to matrices will perform the operation on columns. For example,

sum(rand(10,5))

will yield a row vector with five elements which are the sums of each column.

Moreover

hist(randn(1000,5))

will yield the following multiple histogram with each colour representing a column of the matrix.

Histogram example

Question

Generate a series of random samples of data of increasing size drawn from (i) a uniform distribution and (ii) a normal distribution. Draw histograms of each sample and, by calculating the mean and standard deviation of each sample, verify that these two statistics tend to their theoretical limit values as the size of the sample increases.

Expand for hint
Hint

The mean and variance for a uniform $U(0,1)$ distribution are $\mu=0.5 \,,\, \sigma=1/\sqrt{12}$

The mean and variance for a normal $N(0,1)$ distribution are $\mu=0 \,,\, \sigma=1$.

Expand for solution
Solution

To generate the data for 1,000 samples use

X=rand(1000,1);
Y=randn(1000,1);

To plot the histograms use

figure;
hist(X);
title('Data from Uniform');
figure;
hist(Y);
title('Data from Normal');

To calculate the mean and standard deviation use

mean(X)
std(X)
mean(Y)
std(Y)

which should be close to the theoretical values of $\mu=0.5 \,,\, \sigma=1/\sqrt{12}$ for the $U(0,1)$ data and $\mu=0 \,,\, \sigma=1$ for the $N(0,1)$ data. These values should get closer as the sample size increases.