I am pleased to announce Release 4.8 of the Real Statistics Resource Pack. The new release is now available for free download (Download Resource Pack) for Excel 2007, 2010, 2013 and 2016 (Windows version) environments.

The spreadsheets for all the examples used on the Real Statistics website are available for free download (Download Examples Workbooks) are being updated for compatibility with the new release. These spreadsheets are contained in four Excel files (i.e. workbooks): Examples Workbook Part 1, Examples Workbook Part 2, Multivariate Examples and Time Series Workbook. See Workbook Examples for a description of which examples are contained in which files.

Release 4.8 contains the following new features:

**Shapely-Owen Decomposition**

Adds the following new array function:

**SHAPELY**(Rx, Ry) = a column range which gives the Shapely-Owen decomposition of R-square for the linear regression of Rx (which contains the *x* data values) on Ry (which contains the y data values).

This decomposition shows the relative contribution of each of the independent variables to the regression model.

**Box-Cox Transformation**

Adds the following new functions which carry out Box-Cox transformations

**BOXCOX**(R1, λ): array function which returns a range containing the Box-Cox transformation of the data in range R1 using the given lambda value. If the lambda argument is omitted, then the transformation which best normalizes the data in R1 is used, based on maximizing the log-likelihood function.

**BOXCOXLL**(R1, λ) = log likelihood function of the Box-Cox transformation of the data in R1 using the given lambda value

**BOXCOXLambda**(R1) = the value of lambda which maximizes the log likelihood function of the Box-Cox transformation of the data in R1

**New Support for the Multivariate Normal Distribution**

Adds the following new functions

**BNORMSDIST**(*x, *y*, r, cum*) = the cdf of the standard bivariate normal distribution at *x* and y with correlation coefficient *r* if *cum* = TRUE and the pdf if *cum* = FALSE

**BNORMDIST**(*x*1*, x*2*, m*1*, m*2*, s*1*, s*2*, r, cum*) = the cdf of the standard bivariate normal distribution at at *x*1 and *x*2 with means *m*1 and *m*2, standard deviations *s*1 and *s*2 and correlation coefficient *r* if *cum* = TRUE and the pdf if *cum* = FALSE

**MNORMRAND**(Rm, Rc): generates an array containing a random *k ×* 1 vector from a *k*-dimensional multivariate normal distribution where Rm is the range that contains the *k ×* 1 means vector and Rc is the range that contains the *k ×* *k* covariance matrix

**BNORMRAND**(*m*1*, m*2*, s*1*, s*2*, r*): generates a random 2 *×* 1 vector from a bivariate normal distribution with means *m*1, *m*2, standard deviations *s*1, *s*2 and

correlation coefficient *r*

These last two functions can be used to generate a random vector which comes from a multivariate normal distribution with the specified parameters.

**Mardia’s Test for Multivariate Normality**

Adds the following functions that test whether the data in range R1 is drawn from a multivariate normal distribution

**MSKEWTEST**(R1, *lab*): Mardia’s skewness test for multivariate normality; returns a column range with the values skewness, chi-square statistic, *df* and p-value, plus corrected statistic and p-value for small samples

**MKURTTEST**(R1, *lab*): Mardia’s kurtosis test for multivariate normality; returns a column range with the values kurtosis, z-statistic and p-value

If* lab* = TRUE then an extra column of labels is appended to the results (defaults to FALSE).

**Changes to the Noncentral Chi-square and F Distribution Functions**

The **NCHISQ_DIST**, **NCHISQ_INV**, **NCHISQ_NCP**, **NF_DIST**, **NF_INV** and **NF_NCP** functions have been revised to improve their accuracy especially for large values of the noncentral parameter. Note that the number of terms in the infinite series approximations has been greatly expanded (default is now 1,000 instead of 80). In addition, you can now specify the level of accuracy via a precision argument (default is .000000001).

The same changes have been made to the power and sample size functions **REG_POWER**, **REG_SIZE**, **CHISQ_POWER**, **CHISQ_SIZE, ANOVA1_POWER**, and ** ANOVA1_SIZE**.

The various functions that support the noncentral t distribution and the corresponding power and sample size functions have not changed, except that the default number of terms in the infinite series has been increased from 80 to 120.

These changes are also reflected in the corresponding **Power and Sample Size** data analysis tools.

**Minor Enhancements and Bug Fixes**

- Fixes a bug in the
**KCORREL**function - Added some new arguments to the Mahalanobis distance squared function
**MDISTSq**