degrees of freedom

Statistics

(noun)

any unrestricted variable in a frequency distribution

Related Terms

Physics

(noun)

A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system. The set of all dimensions of a system is known as a phase space.

Related Terms

Examples of degrees of freedom in the following topics:

  • The chi-square distribution and finding areas

    • This distribution has three degrees of freedom, so only the row with 3 degrees of freedom (df) is relevant.
    • (b) 2 degrees of freedom, area above 4.3 shaded.
    • (c) 5 degrees of freedom, area above 5.1 shaded.
    • (e) 4 degrees of freedom, area above 10 shaded.
    • (f) 3 degrees of freedom, area above 9.21 shaded.

  • 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

  • Degrees of Freedom

    • The number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.
    • Each parameter that is fixed during our computations constitutes the loss of a degree of freedom.
    • Put informally, the "interest" in our data is determined by the degrees of freedom.
    • Degrees of freedom can be seen as linking sample size to explanatory power.
    • That smaller dimension is the number of degrees of freedom for error.

  • Notation

    • where df = degrees of freedom depend on how chi-square is being used.
    • The degrees of freedom for the three major uses are each calculated differently. )
    • The random variable for a chi-square distribution with k degrees of freedom is the sum of k independent, squared standard normal variables.

  • 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.
    • What proportion of the t distribution with 18 degrees of freedom falls below -2.10?
    • The t distribution with 18 degrees of freedom.

  • Degrees of Freedom

    • State the general formula for degrees of freedom in terms of the number of values and the number of estimated parameters
    • Since this estimate is based on two independent pieces of information, it has two degrees of freedom.
    • The process of estimating the mean affects our degrees of freedom as shown below.
    • Therefore, the estimate of variance has 2 - 1 = 1 degree of freedom.
    • If we had sampled 12 Martians, then our estimate of variance would have had 11 degrees of freedom.

  • Summary

    • The distribution for the test is the F distribution with 2 different degrees of freedom.
    • A Test of Two Variances hypothesis test determines if two variances are the same.
    • The distribution for the hypothesis test is the F distribution with 2 different degrees of freedom.

  • 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.

  • 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.

  • 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.