What is an inverse cumulative distribution function (ICDF)?

The inverse cumulative distribution function gives the value associated with a specific cumulative probability. Use the inverse CDF to determine the value of the variable associated with a specific probability.

Example of using ICDF to determine warranty periods

For example, an appliance manufacturer investigates failure times for the heating element within its toasters. They want to determine the time by which specific proportions of heating elements will fail so they can set the warranty period. Heating element failure times follow a normal distribution with a mean of 1000 hours and standard deviation of 300 hours. The probability density function (PDF) helps identify regions of higher and lower failure probabilities. The inverse CDF gives the corresponding failure time for each cumulative probability.

Use the inverse CDF to estimate the time by which 5% of the heating elements will fail, times between which 95% of all heating elements will fail, or the time at which only 5% of the heating elements remain. The inverse CDF for specific cumulative probabilities is equal to the failure time at the right side of the shaded area under the PDF curve.

Determine the time at which 5% will fail

  1. Open the inverse cumulative distribution function dialog box.
    • Mac: Statistics > Probability Distributions > Inverse Cumulative Distribution Function
    • PC: STATISTICS > CDF/PDF > Inverse Cumulative Distribution Function
  2. In Form of input, select A single value.
  3. In Value, enter 0.05.
  4. In Distribution, select Normal.
  5. In Mean, enter 1000.
  6. In Standard deviation, enter 300.
  7. Click OK.
Inverse of the Cumulative Probability
P(X ≤ x)
x

The time by which 5% of the heating elements are expected to have failed is the inverse CDF of 0.05 or 507 hours.

Determine times between which 95% will fail

  1. Open the inverse cumulative distribution function dialog box.
    • Mac: Statistics > Probability Distributions > Inverse Cumulative Distribution Function
    • PC: STATISTICS > CDF/PDF > Inverse Cumulative Distribution Function
  2. In Form of input, select A single value.
  3. In Value, enter 0.025.
  4. In Distribution, select Normal.
  5. In Mean, enter 1000.
  6. In Standard deviation, enter 300.
  7. Click OK.

    The time by which 2.5% of the heating elements are expected to have failed is the inverse CDF of 0.025 or 412 hours.

  8. Repeat step 3, but enter 0.975 instead of 0.025. Click OK.
    The time by which 97.5% of the heating elements are expected to have failed is the inverse CDF of 0.975 or 1588 hours.

Therefore, times between which 95% of all heating elements are expected to fail is the inverse CDF of 0.025 and the inverse CDF of 0.975 or 412 hours and 1588 hours.

Determine the time at which only 5% will survive

  1. Open the inverse cumulative distribution function dialog box.
    • Mac: Statistics > Probability Distributions > Inverse Cumulative Distribution Function
    • PC: STATISTICS > CDF/PDF > Inverse Cumulative Distribution Function
  2. In Form of input, select A single value.
  3. In Value, enter 0.95.
  4. In Distribution, select Normal.
  5. In Mean, enter 1000.
  6. In Standard deviation, enter 300.
  7. Click OK.
Inverse of the Cumulative Probability
P(X ≤ x)
x

The time at which only 5% of the heating elements are expected to remain is the inverse CDF of 0.95 or 1493 hours.

Example of using the CDF and the ICDF with the binomial distribution

When you try to determine the inverse cumulative probability of a discrete distribution, the output contains two sets of columns.

Suppose you have the inverse cumulative probability of a proportion, p. The first set of columns in the output lists the largest x such that P(X ≤ x) ≤ p. The second set of columns lists the smallest x such that P(X ≤ x) ≥ p.

Calculate the cumulative probability of a binomial distribution

  1. In worksheet column C1, enter 0 1 2.
    C1
    0
    1
    2
  2. Open the cumulative distribution function dialog box.
    • Mac: Statistics > Probability Distributions > Cumulative Distribution Function
    • PC: STATISTICS > CDF/PDF > Cumulative Distribution Function
  3. In Form of input, select A column of values.
  4. In Values in, select the worksheet column that contains the values that you entered.
  5. In Distribution, select Binomial.
  6. In Number of trials, enter 100.
  7. In Event probability, enter 0.03.
  8. Click OK.

A column of probabilities is stored in the worksheet.

C1 C2
Binomial CDF for C1
0 0.047553
1 0.194622
2 0.419775

You can interpret the output as follows:
  • P(X ≤ 0) = 0.047553. The probability of getting 0 defectives is 5%.
  • P(X ≤ 1) = 0.194622. The probability of getting 0 or 1 defectives is about 19%.
  • P(X ≤ 2) = 0.419775. The probability of getting 0, 1, or 2 defectives is about 42%.

Calculate the inverse cumulative probability of a binomial distribution

Now that you know the cumulative probabilities associated with the number of defectives, calculate the inverse cumulative probability.

Suppose that you want to calculate the number of defectives, x, such that the cumulative probability, p, is 0.50. From the previous results, you know that P(X ≤ 1 ) = 0.194622 and P(X ≤ 2 ) = 0.419775. Because the binomial distribution is a discrete distribution, the number of defectives cannot be between 1 and 2. In other words, you may have 1 defective or 2 defectives, but not 1.4 defectives.

  1. Open the inverse cumulative distribution function dialog box.
    • Mac: Statistics > Probability Distributions > Inverse Cumulative Distribution Function
    • PC: STATISTICS > CDF/PDF > Inverse Cumulative Distribution Function
  2. In Form of input, select A single value.
  3. In Value, enter 0.50.
  4. In Distribution, select Binomial.
  5. In Number of trials, enter 100.
  6. In Event probability, enter 0.03.
  7. Click OK.
Inverse of the Cumulative Probability
x
P(X ≤ x)
x
P(X ≤ x)
Inverse of the Cumulative Probability

The first probability indicates a value of x such that P(X ≤ x) < p and the second probability indicates the smallest x such that P(X ≤ x) ≥ p. In this example, the first probability shows the largest number of defectives, x=2, such that P(X≤2)<0.5 and the 2nd shows the smallest number of defectives, x=3, such that P(X≤3) ≥ 0.5.

Use the ICDF to calculate critical values

You can use Minitab to calculate a critical value for a hypothesis test instead of looking in a table.

Suppose that you perform a chi-square test with an α=0.02 and 12 degrees of freedom. What is the corresponding critical value? An α=0.02 corresponds to a cumulative probability value of 1 - 0.02 = 0.98.

  1. Open the inverse cumulative distribution function dialog box.
    • Mac: Statistics > Probability Distributions > Inverse Cumulative Distribution Function
    • PC: STATISTICS > CDF/PDF > Inverse Cumulative Distribution Function
  2. In Form of input, select A single value.
  3. In Value, enter 0.98.
  4. In Distribution, select Chi-Square.
  5. In Degrees of freedom, enter 12.
  6. Click OK.

Minitab displays the critical value, 24.054. For the chi-square test, if the test statistic is greater than the critical value, you can conclude that there is statistical evidence to reject the null hypothesis.

Note

This example is for a chi-square distribution. However, you can use similar steps for other distributions.

Inverse of the Cumulative Probability
P(X ≤ x)
x
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