Single-Factor ANOVA: Analysis, Assumptions, and Application

Single-Factor ANOVA is used to determine whether three or more populations have equal means.

ANOVA Assumptions:

• The population values are normally distributed.
• The variances for each population are equal.
• The samples are independent.
• The data measurement is interval or ratio level.

ANOVA is an analysis of variance. It needs to have equal variance in order to test the means.

Single-Factor ANOVA: analysis of variance design in which independent samples are obtained from two or more levels of a single factor for the purpose of testing whether the levels have equal means.

Type of data used in ANOVA: ratio or interval data (numbers).

H₀: all means are equal.

H₁: there is a difference in means.

Theory: Practicality.

In Multifactor ANOVA:

• Groups: rows (router 1 – router 4)
• Factors: columns (AT&T, Xfinity, COMCAST)

How to Run an ANOVA in Excel:

1. Open file.
2. Select Data > Data Analysis.
3. Select ANOVA: Single Factor.
4. Define Data Range.
5. Specify Alpha.
6. Indicate output choice.
7. Click OK.

This will give you the summary with the count, sum, average, and variance. It will also include the source of variation between and within groups.

What is Done if There is Significance in Means?

• If there is significance in means, then the null hypothesis needs to be rejected and a Turkey-Kramer Post-Hoc critical value test needs to be conducted.
• This is due to the fact that even though the ANOVA is telling you that there is a difference in means, it doesn’t tell you where that difference is, and with the post hoc you are going to be comparing samples.

Where:

Q1: from the value (k = groups across, DF = rows down). (If DF is between rows, we round to the smaller number)

MSW: mean squares within. Obtained from the ANOVA data.

N1 and n2: counts.

If we reject the null hypothesis, we find the absolute value of difference in means and we set up all the possible scenarios:

1 vs. 2

1 vs. 3

1 vs. 4

2 vs. 3

2 vs. 4

3 vs. 4

After setting up the possible scenarios and finding where the significance is, we perform a T-Test with equal variance between the groups where significance was found.

If F > alpha or P < alpha: we reject the null.

Based on this, the business decision is made.

Single vs. Multiple-Factor ANOVA

A single-factor ANOVA is used when there is just one independent variable (factor) with two or more conditions.

A multiple-factor ANOVA is used when there are two or more independent variables (factors).

Examples:

• Single-factor ANOVA: Aspirin vs. Tylenol vs. Ibuprofen.
• Multi-factor ANOVA:
  • Factor A: aspiring vs. Tylenol
  • Factor B: headache vs. backpain