This lesson describes how to calculate probabilities associated with the difference between two means.

**Difference Between Means**

Often in statistics we’re interested in comparing the difference between two population means. But before we can compare the difference between means, we first need to make sure the following conditions are met:

**Equal Variance:**The two populations need to have roughly the same variance.**Normality:**The sampling distribution of the difference in sample means needs to be approximately normal. This is true if the populations we are studying are both normal or if our sample sizes are sufficiently large (typically n ≥ 30)**Independence:**The two populations need to be independent.

If these conditions are met, we can use the following formulas to quantify the difference between the two means:

The expected difference between all possible sample means: **μ _{d}= **μ

_{1}– μ

_{2}

where μ_{1} is the mean of the first population and μ_{2} is the mean of the second population.

The standard deviation of the difference between sample means: **σ _{d}** = √ σ

_{1}

^{2}/ n

_{1}+ σ

_{2}

^{2}/ n

_{2}

where σ_{1}^{2} is the variance of the first population, n_{1} is the sample size of the first sample, σ_{2}^{2} is the variance of the second population, and n_{2} is the sample size of the second sample.

Let’s walk through an example of how to use these formulas to answer a question about probability.

**Example: Finding the Difference Between Means**

The height of plants in garden A are normally distributed with a mean of 12 inches and a standard deviation of 6 inches. The height of plants in garden B are normally distributed with a mean of 10 inches and a standard deviation of 4 inches.

**Suppose you take a random sample of 80 plants from garden A and 70 plants from garden B. What is the probability that the mean height of plants from garden A will be less than the mean height of plants in garden B?**

**Solution:**

**Step 1: Find the mean difference in the population.**

**μ _{d}= **μ

_{A}– μ

_{B}= 12 – 10 =

**2**

**Step 2: Find the standard deviation of the difference.**

**σ _{d}** = √ σ

_{A}

^{2}/ n

_{A}+ σ

_{B}

^{2}/ n

_{B}

**σ _{d}** = √ 6

^{2}/ 80 + 4

^{2}/ 70

**σ _{d}** =

**0.824**

**Step 3: Find the z-score for when the mean height of A is less than the mean height of B (i.e. when μ _{A} – μ_{B} < 0)**

z = (x – μ)/σ = (0 – 2)/.824 =** -2.43**

**Step 4: Find the cumulative probability associate with the z-score.**

In this case, we want to know the probability that the mean of A is *less *than the mean of B, so we want to find the area to the left of z = -2.43 in the z table, which turns out to be **.0075**:

Thus, the probability that the mean height of plants from garden A will be less than the mean height of plants in garden B is **.0075**.