A **uniform distribution** is a probability distribution in which every value between an interval from *a *to *b *is equally likely to be chosen.

The probability that we will obtain a value between x_{1} and x_{2} on an interval from *a *to *b *can be found using the formula:

P(obtain value between x_{1} and x_{2}) = (x_{2} – x_{1}) / (b – a)

The uniform distribution has the following properties:

- The mean of the distribution is
**μ**= (a + b) / 2 - The variance of the distribution is
**σ**= (b – a)^{2}^{2}/ 12 - The standard deviation of the distribution is
**σ**= √σ^{2}

Let’s walk through some examples to get a better understanding of the uniform distribution.

**Examples Using the Uniform Distribution**

**Example 1: A bus shows up at a bus stop every 20 minutes. If you arrive at the bus stop, what is the probability that the bus will show up in 8 minutes or less?**

**Solution using the formula:**

a = 0 minutes

b = 20 minutes

x_{1} = 0 minutes

x_{2} = 8 minutes

P(bus shows up in 8 minutes or less) = (8 – 0) / (20 – 0) = 8 / 20 = ** 0.4**

**Using the calculator:**

Plug the following numbers into the Uniform Distribution Calculator:

The probability that the bus shows up in 8 minutes or less is 0.4.

**Example 2: A bus shows up at a bus stop every 20 minutes. If you arrive at the bus stop, find the time in which there is a 90% chance that the bus will come by that time.**

**Solution:**

Time = [0.9 * (b – a)] + a = [0.9 * (20 – 0)] + 0 = 18 minutes

**Example 3: The weight of a certain species of frog is uniformly distributed between 15 and 25 grams. If you randomly select a frog, what is the probability that the frog weighs between 17 and 19 grams?**

**Solution using the formula:**

a = 15 grams

b = 25 grams

x_{1} = 17 grams

x_{2} = 19 grams

P(weighs between 17 and 19 grams) = (19 – 17) / (25 – 15) = 2 / 10 = ** 0.2**

**Using the calculator:**

Plug the following numbers into the Uniform Distribution Calculator:

The probability that a randomly selected frog weighs between 17 and 19 grams is 0.2.

**Example 4: The weight of a certain species of frog is uniformly distributed between 15 and 25 grams. Find the 80th percentile for a frog’s weight.**

**Solution:**

Weight = [0.8 * (b – a)] + a = [0.8 * (25 – 15)] + 15 = 23 grams

**Example 5: The length of an NBA game is uniformly distributed between 120 and 170 minutes. What is the probability that a randomly selected NBA game lasts more than 2 and a half hours?**

**Solution using the formula:**

a = 2 and a half hours = 150 minutes

b = 170 minutes

x_{1} = 120 minutes

x_{2} = 170 minutes

P(game lasts more than 2 and a half hours) = (170 – 150) / (170 – 120) = 20 / 50 = ** 0.4**

**Using the calculator:**

Plug the following numbers into the Uniform Distribution Calculator:

**Example 6: The length of an NBA game is uniformly distributed between 120 and 170 minutes. Find the length of a game in which only 5% of games last longer.**

**Solution:**

Length = [0.95 * (b – a)] + a = [0.95 * (170 – 120)] + 120 = 167.5 minutes