17.1.2.1 The Statistics on Rows Dialog Box

Supporting Information

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 Button Group List Move Up.png, Move Down button Button Group List Move Down.png, Remove button Button Group List Remove.png, Select All button Button Group List Select All.png, Select button Button Group List Add.png in toolbar Group List Toolbar.png.

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 atx_{(np)}\,\!

y=(1-g)x_{(j)}+gx_{(j+1)}\,\!

where x_{(0)} is taken to bex_{(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.

Output

Report Tables Specifies the destination of report worksheet tables
Book
Specifies the destination workbook.
  • <none>: Do not output report worksheet tables.
  • <source>: The source data workbook.
  • <new>: A new workbook.
  • <existing>: A specified existing workbook
BookName
Enter the name of the workbook.
Sheet
The target worksheet.
SheetName
Name of the target worksheet.
Column
  • <new> Output stats to appended columns.
  • <next to source> 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:
  • <none>: Do not output to a Notes window.
  • <new>: Output to a new Notes window. Specify a name for the Notes window.