# 17.1.2.1 The Statistics on Rows Dialog Box

## Input

Input Data Specify the data range to be performed: Data Range The input data range For help with range controls, see: Specifying Your Input Data Group Multiple column label rows contains grouping information can be inserted into the Group box. Different grouping values indicate the data in the corresponding cells are from different groups. You can add, remove, order grouping rows via controlling buttons: Move Up button , Move Down button , Remove button , Select All button , Select button in toolbar .

## Quantities

### Moments

Let $x_i$ be the ith sample and $w_i$ be the ith weight.
N Total Total number of data points, denoted by n Number of missing values The mean (average) score $\bar{x}=\frac 1n\sum_{i=1}^n x_i$. $s=\sqrt{\sum_{i=1}^n (x_i-\bar{x})^2/d}$ where $d=n-1 \,$ Note: In OriginPro, $d$ has another option, defined in the Variance Divisor of Moment branch. Standard error of mean: $\frac S{\sqrt{n}}$ Lower limit of the 95% confidence interval of mean $\bar{x}-t_{(1-\alpha/2)}\frac{s}{\sqrt{n}}$ where $t_{(1-\alpha/2)}$ is the $1-\alpha/2$ critical value of the Student's t-statistic with n-1 degrees of freedom Upper limit of the 95% confidence interval of mean $\bar{x}+t_{(1-\alpha/2)}\frac{s}{\sqrt{n}}$ where $t_{(1-\alpha/2)}$ is the $1-\alpha/2$ critical value of the Student's t-statistic with n-1 degrees of freedom $s^2$ $\sum_{i=1}^n x_i$. Skewness measures the degree of asymmetry of a distribution. It is defined as $\gamma_1=\frac n{(n-1)(n-2)}\sum_{i=1}^n (\frac{x_i-\bar{x}}s)^3 ,\mbox{for DF}$ $\gamma_1=\frac 1n\sum_{i=1}^n (\frac{x_i-\bar{x}}s)^3,\mbox{for N}$ $\gamma_1=\frac 1n\sum_{i=1}^n (\frac{x_i-\bar{x}}s)^3,\mbox{for WVR}$ Kurtosis depicts the degree of peakedness of a distribution. $\gamma_2=\frac{n(n+1)}{(n-1)(n-2)(n-3)}\sum_{i=1}^n (\frac{x_i-\bar{x}}s)^4-\frac{3(n-1)^2}{(n-2)(n-3)},\mbox{for DF}$ $\gamma_2=\frac 1n\sum_{i=1}^n (\frac{x_i-\bar{x}}s)^4 -3,\mbox{for N}$ $\gamma_2=\frac 1n\sum_{i=1}^n (\frac{x_i-\bar{x}}s)^4 -3,\mbox{for WVR}$ $\sum_{i=1}^n x_i^2$ $\sum_{i=1}^n (x_i-\bar{x})^2$ $\frac s{\bar{x}}$ $\frac{\sum_{i=1}^n |x_i-\bar{x}|}n$ Standard deviation times 2. $2s \,$ Standard deviation times 3. $3s \,$ $\bar{x}_g=\left( \prod_{i=1}^n x_i\right) ^{\frac 1n}$ The geometric standard deviation $e^{std(\log x_i)}$ Where std is the unweighted sample standard deviation. Note: Weights are ignored for the geometric standard deviation. The mode is the element that appears most often in the data range. If multiple modes are found, the smallest will be chosen. harmonic mean (sometimes called the subcontrary mean) without weight: $\frac n{\frac 1{x_1} + \frac 1{x_2} + ... + \frac 1{x_n}}=\left(\frac {\sum_{i=1}^n (x_i)^{-1}}n\right)^{-1}$ with weight: $\frac {\sum_{i=1}^n w_i}{\sum_{i=1}^n \frac {w_i}{x_i}}=\left(\frac {\sum_{i=1}^n w_i x_i^{-1}}{\sum_{i=1}^n w_i}\right)^{-1}$ if any $x_i$ or weight is negative, return missing; if any $x_i$ or weight is 0, return 0.

### Quantiles

Quantiles are values from the data, below which is a given proportion of the data points in a given set. For example, 25% of data points in any set of data lay below the first quartile, and 50% of data points in a set lay below the second quartile, or median.

Sort the input dataset in ascending order. Let $x_{(i)}\,\!$ be the ith element of the reordered dataset

Minimum $x_{(i)}\,\!$ The index number of Minimum in the original (input) dataset. First (25%) quantile, Q1. See Interpolation of quantiles for computational methods. Median or second (50%) quantile, Q2. See Interpolation of quantiles for computational methods. Third (75%) quantile, Q3. See Interpolation of quantiles for computational methods. $x_{(n)}\,\!$ The index number of Maximum in the original (input) dataset. $Q_3-Q_1\,$ Maximum - Minimum Request computation of custom percentiles. This option is only available when Custom Percentile(s) is checked. Percentiles are computed for all the values listed. For a univariate data set X1, X2, ..., Xn, the MAD is defined as the median of the absolute deviations from the data's median: $MAD = median(|{X_i} - median(X)|)\,$ that is, starting with the residuals (deviations) from the data's median, the MAD is the median of their absolute values. $(MAD/norminv(0.75))/Median\,$

## Computation Control

### Variance Divisor of Moment

Controls computation of variance divisor d
DF Degree of freedom $d=n-1\,\!$ Number of non-missing observations. $d=n\,\!$

### Interpolation of quantiles

This option decides the methods for calculating Q1, Q2, and Q3.
Let the ith percentile be y, set $p=i/100$, and let

$\begin{cases} (n+1)p=j+g, & \mbox{for Weighted Average Right}\\ np=j+g, & \mbox{for other methods} \end{cases}$

where j is the integer part of np, and g is the fractional part of np, then different methods define the $i^{th}\,\!$ percentile, y, as described by the following:
Empirical Distribution with Averaging $y=\begin{cases} \frac{1}{2}(x_{(j)}+x_{(j+1)}), & \mbox{if }g=0\\ x_{(j+1)}, & \mbox{if }g>0 \end{cases}$ Observation numbered closest to np $y=\begin{cases} x_{(k)}, & \mbox{if }g\ne \frac{1}{2}\\ x_{(j)}, & \mbox{if }g=\frac{1}{2} \mbox{ and } j\mbox{ is even} \\ x_{(j+1)}, & \mbox{if }g=\frac{1}{2} \mbox{ and } j\mbox{ is odd} \end{cases}$ where k is the integer part of $np+\frac{1}{2}$ $y=\begin{cases} x_{(j)}, & \mbox{if }g=0 \\ x_{(j+1)}, & \mbox{if }g>0 \end{cases}$ weighted average aimed at $x_{(n+1)+p)}\,\!$ $y=(1-g)x_{(j)}+gx_{(j+1)}\,\!$ where $x_{(n+1)}\,\!$is taken to be $x_{(n)}\,\!$ weighted average aimed at$x_{(np)}\,\!$ $y=(1-g)x_{(j)}+gx_{(j+1)}\,\!$ where $x_{(0)}$ is taken to be$x_{(1)}$ Let: $m=\begin{cases} \frac{n}{2},& \mbox{if }n\mbox{ is even}\\ \frac{n+1}{2},& \mbox{if }n\mbox{ is odd} \end{cases}$ $k=\begin{cases} \frac{m}{2},& \mbox{if }m\mbox{ is even}\\ \frac{m+1}{2},& \mbox{if }m\mbox{ is odd} \end{cases}$ Then we have: $Minimum+x_{(1)}\,\!$ $Q_1=\begin{cases} x_{(k)},& \mbox{if }m\mbox{ is odd}\\ \frac{1}{2}(x_{(k)}+x_{(k+1)}), & \mbox{if }m\mbox{ is even} \end{cases}$ $Q_2=\begin{cases} x_{(m)},& \mbox{if }n\mbox{ is odd}\\ \frac{1}{2}(x_{(m)}+x_{(m+1)}), & \mbox{if }m\mbox{ is even} \end{cases}$ $Q_3=\begin{cases} x_{(n-k-1)},& \mbox{if }n\mbox{ is odd}\\ \frac{1}{2}(x_{(n-k)}+x_{(mn-k+1)}), & \mbox{if }m\mbox{ is even} \end{cases}$ $Maximum=x_{(n)}\,\!$ Note: if this method is selected, only quartiles will be computed. Custom percentiles are disabled.

## Output

Report Tables Specifies the destination of report worksheet tables Book Specifies the destination workbook. : Do not output report worksheet tables. : The source data workbook. : A new workbook. : A specified existing workbook BookName Enter the name of the workbook. Sheet The target worksheet. SheetName Name of the target worksheet. Column Output stats to appended columns. Output stats to appended or inserted columns. Results Log Output the report to the Results Log Script Window Output the report to the Script Window Notes Window Specify the destination Notes window: : Do not output to a Notes window. : Output to a new Notes window. Specify a name for the Notes window.