# Introductory Statistics

Mathematics and Statistics## Facts About the Chi-Square Distribution

The notation for the chi-square distribution is:

where *df* = degrees of freedom which depends 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 *χ ^{2}* distribution, the population mean is

*μ*=

*df*and the population standard deviation is $\sigma =\sqrt{2(df)}$.

The random variable is shown as *χ ^{2}*, but may be any upper case letter.

The random variable for a chi-square distribution with *k* degrees of freedom is the sum of *k* independent, squared standard normal variables.

*χ*^{2} = (*Z*_{1})^{2} + (*Z*_{2})^{2} + ... + (*Z*_{k})^{2}

- The curve is nonsymmetrical and skewed to the right.
- There is a different chi-square curve for each
*df*. - The test statistic for any test is always greater than or equal to zero.
- When
*df*> 90, the chi-square curve approximates the normal distribution. For*X*~ ${\chi}_{\mathrm{1,000}}^{2}$ the mean,*μ*=*df*= 1,000 and the standard deviation,*σ*= $\sqrt{2(\mathrm{1,000})}$ = 44.7. Therefore,*X*~*N*(1,000, 44.7), approximately. - The mean,
*μ*, is located just to the right of the peak.

# References

Data from *Parade Magazine*.

“HIV/AIDS Epidemiology Santa Clara County.”Santa Clara County Public Health Department, May 2011.

# Chapter Review

The chi-square distribution is a useful tool for assessment in a series of problem categories. These problem categories include primarily (i) whether a data set fits a particular distribution, (ii) whether the distributions of two populations are the same, (iii) whether two events might be independent, and (iv) whether there is a different variability than expected within a population.

An important parameter in a chi-square distribution is the degrees of freedom *df* in a given problem. The random variable in the chi-square distribution is the sum of squares of *df* standard normal variables, which must be independent. The key characteristics of the chi-square distribution also depend directly on the degrees of freedom.

The chi-square distribution curve is skewed to the right, and its shape depends on the degrees of freedom *df*. For *df* > 90, the curve approximates the normal distribution. Test statistics based on the chi-square distribution are always greater than or equal to zero. Such application tests are almost always right-tailed tests.

# Formula Review

*χ*^{2} = (*Z*_{1})^{2} + (*Z*_{2})^{2} + … (*Z _{df}*)

^{2}chi-square distribution random variable

*μ _{χ2}* =

*df*chi-square distribution population mean

${\sigma}_{{\chi}^{2}}\text{=}\sqrt{2\left(df\right)}$ Chi-Square distribution population standard deviation

If the number of degrees of freedom for a chi-square distribution is 25, what is the population mean and standard deviation?

mean = 25 and standard deviation = 7.0711

If *df* > 90, the distribution is _____________. If *df* = 15, the distribution is ________________.

When does the chi-square curve approximate a normal distribution?

when the number of degrees of freedom is greater than 90

Where is *μ* located on a chi-square curve?

Is it more likely the *df* is 90, 20, or two in the graph?

*df* = 2

# Homework

*Decide whether the following statements are true or false.*

As the number of degrees of freedom increases, the graph of the chi-square distribution looks more and more symmetrical.

true

The standard deviation of the chi-square distribution is twice the mean.

The mean and the median of the chi-square distribution are the same if *df* = 24.

false

- Introductory Statistics
- Preface
- Sampling and Data
- Descriptive Statistics
- Introduction
- Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs
- Histograms, Frequency Polygons, and Time Series Graphs
- Measures of the Location of the Data
- Box Plots
- Measures of the Center of the Data
- Skewness and the Mean, Median, and Mode
- Measures of the Spread of the Data
- Descriptive Statistics

- Probability Topics
- Discrete Random Variables
- Introduction
- Probability Distribution Function (PDF) for a Discrete Random Variable
- Mean or Expected Value and Standard Deviation
- Binomial Distribution
- Geometric Distribution
- Hypergeometric Distribution
- Poisson Distribution
- Discrete Distribution (Playing Card Experiment)
- Discrete Distribution (Lucky Dice Experiment)

- Continuous Random Variables
- The Normal Distribution
- The Central Limit Theorem
- Confidence Intervals
- Hypothesis Testing with One Sample
- Hypothesis Testing with Two Samples
- The Chi-Square Distribution
- Linear Regression and Correlation
- F Distribution and One-Way ANOVA
- Appendix A: Review Exercises (Ch 3-13)
- Appendix B: Practice Tests (1-4) and Final Exams
- Appendix C: Data Sets
- Appendix D: Group and Partner Projects
- Appendix E: Solution Sheets
- Appendix F: Mathematical Phrases, Symbols, and Formulas
- Appendix G: Notes for the TI-83, 83+, 84, 84+ Calculators
- Appendix H: Tables