17.1.2.1 The Statistics on Rows Dialog Box

1 Supporting Information
2 Input
3 Quantities
- 3.1 Moments
- 3.2 Quantiles
4 Computation Control
- 4.1 Variance Divisor of Moment
- 4.2 Interpolation of quantiles
5 Output

Supporting Information

Input

Input Data

Specify the data range to be performed:

Data Range

The input data range. Support for multiple sheets as input (e.g. [Book1](1:2)!A[2]:B[8]).

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
N Missing	Number of missing values
Mean	The mean (average) score $\bar{x}=\frac 1n\sum_{i=1}^n x_i$ .
Standard deviation	$s=\sqrt{\sum_{i=1}^n (x_i-\bar{x})^2/d}$ where $d=n-1 \,$ Note: In OriginPro, $d$ has another option, defined in the Variance Divisor of Moment branch.
SE of Mean	Standard error of mean: $\frac S{\sqrt{n}}$
Lower 95% CI of Mean	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 95% CI of Mean	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
Variance	$s^2$
Sum	$\sum_{i=1}^n x_i$ .
Skewness	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	Kurtosis depicts the degree of peakedness of a distribution. $\gamma_2=\frac{n(n+1)}{(n-1)(n-2)(n-3)}\sum_{i=1}^n (\frac{x_i-\bar{x}}s)^4-\frac{3(n-1)^2}{(n-2)(n-3)},\mbox{for DF}$ $\gamma_2=\frac 1n\sum_{i=1}^n (\frac{x_i-\bar{x}}s)^4 -3,\mbox{for N}$ $\gamma_2=\frac 1n\sum_{i=1}^n (\frac{x_i-\bar{x}}s)^4 -3,\mbox{for WVR}$
Uncorrected Sum of Squares	$\sum_{i=1}^n x_i^2$
Corrected Sum of Squares	$\sum_{i=1}^n (x_i-\bar{x})^2$
Coefficient of Variance	$\frac s{\bar{x}}$
Mean absolute Deviation	$\frac{\sum_{i=1}^n \|x_i-\bar{x}\|}n$
SD times 2	Standard deviation times 2. $2s \,$
SD times 3	Standard deviation times 3. $3s \,$
Geometric Mean	$\bar{x}_g=\left( \prod_{i=1}^n x_i\right) ^{\frac 1n}$
Geometric SD	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.
Mode	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	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	$x_{(i)}\,\!$
Index of Minimum	The index number of Minimum in the original (input) dataset.
1st Quartile (Q1)	First (25%) quantile, Q1. See Interpolation of quantiles for computational methods.
Median	Median or second (50%) quantile, Q2. See Interpolation of quantiles for computational methods.
3rd Quartile (Q3)	Third (75%) quantile, Q3. See Interpolation of quantiles for computational methods.
Maximum	$x_{(n)}\,\!$
Index of Maximum	The index number of Maximum in the original (input) dataset.
Interquartile Range (Q3-Q1)	$Q_3-Q_1\,$
Range (Maximum-Minimum)	Maximum - Minimum
Custom Percentile(s)	Request computation of custom percentiles.
Percentile list	This option is only available when Custom Percentile(s) is checked. Percentiles are computed for all the values listed.
Median Absolute Deviation	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.
Robust Coefficient of Variation	$(MAD/norminv(0.75))/Median\,$

Computation Control

Variance Divisor of Moment

Controls computation of variance divisor d

DF	Degree of freedom $d=n-1\,\!$
N	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	$y=\begin{cases} \frac{1}{2}(x_{(j)}+x_{(j+1)}), & \mbox{if }g=0\\ x_{(j+1)}, & \mbox{if }g>0 \end{cases}$
Nearest Neighbor	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 $np+\frac{1}{2}$
Empirical Distribution	$y=\begin{cases} x_{(j)}, & \mbox{if }g=0 \\ x_{(j+1)}, & \mbox{if }g>0 \end{cases}$
Weighted Average Right	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 Left	weighted average aimed at $x_{(np)}\,\!$ $y=(1-g)x_{(j)}+gx_{(j+1)}\,\!$ where $x_{(0)}$ is taken to be $x_{(1)}$
Tukey Hinges	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.