degrees of freedom (df)

(noun)

The number of objects in a sample that are free to vary.

Related Terms

Examples of degrees of freedom (df) in the following topics:

  • Practice: One-Way ANOVA

    • Suppose that the following data are randomly collected from five teenagers in each region of the country.
    • State the decisions and conclusions (in complete sentences) for the following preconceived levels of α .

  • The chi-square distribution and finding areas

    • The chi-square distribution has just one parameter called degrees of freedom (df), which influences the shape, center, and spread of the distribution.
    • This distribution has three degrees of freedom, so only the row with 3 degrees of freedom (df) is relevant.
    • (c) The distribution is very strongly skewed for df = 2, and then the distributions become more symmetric for the larger degrees of freedom df = 4 and df = 9.
    • (b) 2 degrees of freedom, area above 4.3 shaded.
    • (e) 4 degrees of freedom, area above 10 shaded.

  • Introducing the t distribution

    • The degrees of freedom (df) describe the precise form of the bell-shaped t distribution.
    • The degrees of freedom describe the shape of the t distribution.
    • In Section 5.3.3, we relate degrees of freedom to sample size.
    • We identify the row in the t table using the degrees of freedom: df = 20.
    • The t distribution with 18 degrees of freedom.

  • Notation

    • where df = degrees of freedom depend on how chi-square is being used.
    • (If you want to practice calculating chi-square probabilities then use df = n−1.
    • The degrees of freedom for the three major uses are each calculated differently. )
    • For the χ2 distribution, the population mean is µ = df and the population standard deviation is $\sigma = \sqrt{2 \cdot df}$.
    • The random variable for a chi-square distribution with k degrees of freedom is the sum of k independent, squared standard normal variables.

  • The t distribution as a solution to the standard error problem

    • Use the t distribution for inference of the sample mean when observations are independent and nearly normal.
    • Independence of observations.
    • We collect a simple random sample from less than 10% of the population, or if it was an experiment or random process, we carefully check to the best of our abilities that the observations were independent.
    • When examining a sample mean and estimated standard error from a sample of n independent and nearly normal observations, we use a t distribution with n − 1 degrees of freedom (df).
    • For example, if the sample size was 19, then we would use the t distribution with df = 19 − 1 = 18 degrees of freedom and proceed exactly as we did in Chapter 4, except that now we use the t table.

  • One sample t confidence intervals

    • The value $t^*_{df}$is a cutoff we obtain based on the confidence level and the t distribution with df degrees of freedom.
    • Before determining this cutoff, we will first need the degrees of freedom.
    • If the sample has n observations and we are examining a single mean, then we use the t distribution with df = n−1 degrees of freedom.
    • In our current example, we should use the t distribution with df = 19−1 = 18 degrees of freedom.
    • Degrees of freedom: df = n−1 = 14.

  • The chi-square test for two-way tables

    • However, the degrees of freedom are computed a little differently for a two-way table.
    • For two way tables, the degrees of freedom is equal to
    • df = (number of rows minus 1) × (number of columns minus 1)
    • Looking in Appendix B.3 on page 412, we examine the row corresponding to 2 degrees of freedom.
    • Because there are 2 rows and 3 columns, the degrees of freedom for the test is df = (2 − 1) × (3 − 1) = 2.

  • Chi Square Distribution

    • Describe how the shape of the Chi Square distribution changes as its degrees of freedom increase
    • The mean of a Chi Square distribution is its degrees of freedom.
    • Chi Square distributions are positively skewed, with the degree of skew decreasing with increasing degrees of freedom.
    • Notice how the skew decreases as the degrees of freedom increase.
    • Chi Square distributions with 2, 4, and 6 degrees of freedom

  • t Distribution

    • State how the difference between the shape of the t distribution and normal distribution is affected by the degrees of freedom
    • The t distribution is very similar to the normal distribution when the estimate of variance is based on many degrees of freedom, but has relatively more scores in its tails when there are fewer degrees of freedom.
    • Figure 1 shows t distributions with 2, 4, and 10 degrees of freedom and the standard normal distribution.
    • The t distribution approaches the normal distribution as the degrees of freedom increase.
    • Notice that with few degrees of freedom, the values of t are much higher than the corresponding values for a normal distribution and that the difference decreases as the degrees of freedom increase.

  • Summary of Formulas

    • Use goodness-of-fit to test whether a data set fits a particular probability distribution.
    • The degrees of freedom are number of cells or categories - 1.
    • The degrees of freedom are equal to (number of columns - 1)(number of rows - 1).
    • The degrees of freedom are equal to number of columns - 1.
    • The degrees of freedom are the number of samples - 1.