These tolerance limits, taken from the estimated interval, are limits within which a stated proportion of the population is expected to occur. The function normtol.int from the tolerance package can be used to calculate a tolerance interval for data from a normal distribution.

The function arguments include the data itself in a vector denoted x. The confidence level associated with the tolerance interval is specified by alpha, where alpha is the difference between 100% and the confidence level – alpha is 0.05 for 95% confidence. The argument P is the proportion of the data to be included in the tolerance interval. The side argument determines whether a one-sided or two-sided interval is required.

Consider a simulated set of data from a manufacturing process loaded into R, stored as vector object obs, as follows:

obs = c(102.17, 102.45, 106.23, 98.16, 100.82, 101.40, 90.51, 102.51, 97.93,
  96.98, 101.74, 104.34, 103.50, 94.72, 102.80, 103.92, 97.43, 102.76, 100.03,
  107.12, 104.96, 105.32, 87.06, 97.89, 100.23)

A 95% tolerance interval for 90% of data of this type, based on the 25 observations above is created with this code:

> normtol.int(x = obs, alpha = 0.05, P = 0.90, side = 2)
  alpha   P    x.bar 2-sided.lower 2-sided.upper
1  0.05 0.9 100.5192      90.07606      110.9623

The alpha and P are as noted above and the average of the data is reported along with the lower and upper tolerance intervals in this case as we asked for a two-sided interval. This can be easily changed to cover 95% rather than 90% of the data:

> normtol.int(x = obs, alpha = 0.05, P = 0.95, side = 2)
  alpha    P    x.bar 2-sided.lower 2-sided.upper
1  0.05 0.95 100.5192      88.07543      112.9630

The package tolerance can create intervals for other data distributions.

To illustrate this we could the standard normal distribution which has zero mean and variance of one and the cumulative density function has the familiar S-shape. To plot the distribution on a graph we first create a variable to store the values for the distribution, which we set to be a sequence ranging from -4 to +4 and save the data to a variable tempX so that it can be used in the plot function:

tempX = seq(-4, 4, 0.1)

The next step is to call the plot function and we provide a list of X and Y values that we want to plot against each other. In this case we have already defined the X values so we use the pnorm function to calculate the cumulative values at each of the X values that we have specified. We also set the text for the title and the two axis using the arguments main, xlab and ylab. We use the expression function to create a text string with Mathematical characters in it. The mu and sigma are converted to the corresponding greek letters. Lastly the option type = “l” is used to get the plot function to draw lines rather than symbols. Our final function call is:

plot(tempX, pnorm(tempX, mean=0, sd=1), xlab="X Values",
  ylab="Cumulative Probability", 
  main = expression(paste("Normal Distribution: ", mu, " = 0, ",
    sigma, " = 1")), type="l")

We add a horizontal grey line at the bottom of the graph using the abline function:

abline(h=0, col="gray")

The graph that is produced looks like this:

Plot of the Cumulative Standard Normal Distribution

We can use this approach to visualise the density or cumulative density functions of any distribution that is available in R.