# THE F DISTRIBUTION and F-RATIO IN STATISTICS

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The distribution used for the hypothesis test is a new one. It is called the F distribution, named after Sir Ronald Fisher, an English statistician.

The F statistic is a ratio (a fraction). There are two sets of degrees of freedom; one for the numerator and one for the denominator.

For example, if F follows an F distribution and the number of degrees of freedom for the numerator is four, and the number of degrees of freedom for the denominator is ten, then F ~ F4,10.

To calculate the F ratio, two estimates of the variance are made.

1. Variance between samples: An estimate of σ2
that is the variance of the sample means multiplied by n (when the
sample sizes are the same.).

If the samples are different sizes, the variance between samples is weighted to account for the different sample sizes.

The variance is also called variation due to treatment or explained variation.

2. Variance within samples: An estimate of σthat is the average of the sample variances (also known as a pooled
variance).

When the sample sizes are different, the variance within samples is weighted. The variance is also called the variation due to error or unexplained variation.

• SSbetween = the sum of squares that represents the variation among the different samples

• SSwithin = the sum of squares that represents the variation within samples that is due to chance.

To find a “sum of squares” means to add together squared quantities that, in some cases, may be weighted.

We used sum of squares to calculate the sample variance and the sample standard deviation in Descriptive Statistics.

MS means ” mean square.” MSbetween is the variance between groups, and MSwithin is the variance within groups.

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### Z-Scores

If X is a normally distributed random variable and X ~ N(μ, σ), then the z-score is:

z =x – μ/σ
The z-score tells you how many standard deviations the value x is above (to the right of) or below (to the left of) the mean, μ.

Values of x that are larger than the mean have positive z-scores, and values of x that are smaller than the mean have negative z-scores.

If x equals the mean, then x has a z-score of zero.

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