17.1.1.1 The Statistics on Columns Dialog Box


Supporting Information

Input

Exclude Empty Dataset

Check this check box to exclude empty dataset in calculation.

Input Data

Specify the input data mode, indexed or raw.

Independent Columns

Each column will be treated as separate dataset. Statistical operations will be performed on each, independently.

Combined as Single Dataset

All columns will be treated as a whole.

Range1

Specify the data range to be performed:

Data Range
The input data range
Group
Multiple grouping columns 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 columns 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.
The grouping columns are set as categorical if most of the column values are text. So you can easily reorder the output columns.
Weighting Range
The range that contains weighting information. The value in each cell specifies the weight of the corresponding data.

Quantities

Moments

Let x_i\, be the i\,th sample and w_i\, be the i\,th 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 1w\sum_{i=1}^n x_iw_i. If there is no WEIGHT variable, the formula reduces to \frac 1n\sum_{i=1}^n x_i.

Standard deviation

s=\sqrt{\sum_{i=1}^n w_i(x_i-\bar{x})^2/d}

where d=n-1 \,

Note: In OriginPro, d has 4 more options, which are defined in the Variance Divisor of Moment branch.

SE of Mean Standard error of mean:

\frac s{\sqrt{w}}

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_iw_i. If there is no WEIGHT variable, the formula reduces to \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 w_i^{\frac 32}(\frac{x_i-\bar{x}}s)^3 ,\mbox{for DF}

\gamma_1=\frac 1n\sum_{i=1}^n w_i^{\frac 32}(\frac{x_i-\bar{x}}s)^3,\mbox{for N}

\gamma_1=\frac 1d\sum_{i=1}^n w_i^{\frac 32}(\frac{x_i-\bar{x}}s)^3,\mbox{for WVR}

Note: When the WDF or WS methods are chosen, skewness is returned as a missing value.

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 w_i^2(\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 w_i^2(\frac{x_i-\bar{x}}s)^4 -3,\mbox{for N}

\gamma_2=\frac 1d\sum_{i=1}^n w_i^2(\frac{x_i-\bar{x}}s)^4 -3,\mbox{for WVR}

Note: When the WDF or WS methods are chosen, kurtosis is returned as a missing value.

Uncorrected Sum of Squares

\sum_{i=1}^n w_ix_i^2

Corrected Sum of Squares

\sum_{i=1}^n w_i(x_i-\bar{x})^2

Coefficient of Variance

\frac s{\bar{x}}

Mean absolute Deviation

\frac{ \sum_{i=1}^n w_i|x_i-\bar{x}|}w

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}

Note:: Weights are ignored for the geometric mean.

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.

Sum of Weights

w=\sum_{i=1}^n w_i

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 i\,th element of the reordered dataset

Minimum

x_{(1)}\,

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. 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\,

Extreme Values

Return extreme values. Extreme values are the lst highest and the lst lowest values.
l = 
\begin{cases} 
 5,& \mbox{if }\ n\geq 10 \\
  n/2, & \mbox{otherwise }
\end{cases}

where n is the length of the dataset.

Computation Control

Weight Method

Choose weighting methods for input data.

Direct Weighting

w_{i}=c_{i}\,\!, where c_{i}\,\! is the ith value of weighting dataset.

Instrumental

w_{i}=\frac 1{\sigma _{i}^2}\,\!, where \sigma_{i}\,\! is the value in a designated error bar column.

Statistical

w_{i}=\frac 1{x_{i}}\,\!, where x_{i}\,\! is the input data.

Variance Divisor of Moment

Controls computation of variance divisor d.

DF Degrees of freedom

d=n-1\,

N Number of non-missing observations

d=n\,

WDF Sum of weights DF

d=w-1\,

WS Sum of weights

d=w\,

WVR d=w-\sum_{i=1}^n w_i^2/w

Interpolation of quantiles

Methods for calculating Q1, Q2, and Q3:

Let the i\,th percentile be y, setp=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 12(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\neq \frac 12 \\
  x_j,  & \mbox{if }\ g=\frac 12 \mbox{ and j is even} \\
  x_{(j+1)},& \mbox{if }\ g=\frac 12 \mbox{ and j is odd}
\end{cases} where k is the integer part of np+\frac 12\,

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 n2,& \mbox{if n is even} \\
  \frac{(n+1)}2,& \mbox{if n is odd}
\end{cases}

k = 
\begin{cases} 
 \frac m2,& \mbox{if m is even} \\
  \frac{(m+1)}2,& \mbox{if m is odd}
\end{cases}

Then we have:

Minimun=x_{(1)}\,


Q_1= 
\begin{cases} 
 x_k,& \mbox{if m is odd} \\
  \frac 12(x_{(k)}+x_{(k+1)}),& \mbox{if m is even}
\end{cases} Q_2= 
\begin{cases} 
 x_m,& \mbox{if n is odd} \\
  \frac 12(x_{(m)}+x_{(m+1)}),& \mbox{if n is even}
\end{cases}

Q_3= 
\begin{cases} 
 x_{(n-k-1)},& \mbox{if m is odd} \\
  \frac 12(x_{(n-k)}+x_{(n-k+1)}),& \mbox{if m is even}
\end{cases}

Maximum=x_{(n)}\,

Note: if weights are specified, Weighted Percentiles are calculated. The pth weighted percentile y is computed from the Empirical Distribution Function with Averaging: y= 
\begin{cases} 
  \frac 12(x_{(i)}+x_{(i+1)}),& \mbox{if }  \sum_{j=1}^i w_j=pw \\
  x_{(i+1)},& \mbox{if }  \sum_{j=1}^{i} w_j<pw<\sum_{j=1}^{i+1}w_j\\
  x_{(1)},& \mbox{if } \ pw<w_1 \\
  x_{(n)},& \mbox{if } \ pw<w_n \\
\end{cases}

Output

Graph Control arrangement of resulting plots.
Arrange Graphs into Columns
Specify number of columns to arrange output graphs.
Arrange Plots of Same Type in One Graph
Check to plot all graphs of same type in one graph window.
Dataset Identifier Choose an identifier for the source datasets.
Identifier
Select from the list:
  • Range
Use the range syntax.
  • Book Name
Use the workbook long name.
  • Sheet Name
Use the worksheet name.
  • Name
Use column Long Name if it exists, otherwise Short Name.
  • Short Name
Use column Short Name.
  • Long Name
Use column Long Name.
  • Units
Use column Units.
  • Comments
Use column Comments.
  • <Custom>
Use custom formats to define a data identifier. For usage information, please refer to Advanced legend text customizations.
Show Identifier in Flat Sheet
Use the dataset identifier in resulting flat sheet(s).
Report Tables Destination for report worksheet tables.
Book
The target workbook.
  • <none>: Do not output report worksheet tables.
  • <source>: The source data workbook.
  • <new>: A new workbook.
  • <existing>: An existing workbook.
BookName
The target workbook. Must be source (uneditable), new or existing (editable), otherwise blank.
Sheet
The target worksheet, always <new>.
SheetName
Name of the target sheet.
Results Log
Output the report to the Results Log.
Script Window
Output the report to the Script Window.
Notes Window
Specify the target Notes window:
  • <none>: Do not output to a Notes window.
  • <new>: Output to a new Notes window.
Quantities Specifies the destination of quantities
Book
Specifies the destination workbook
  • <none>: Do not output quantities
  • <source>: The source data workbook
  • <report>: The Report Tables workbook
  • <new>: A new workbook.
  • <existing>: A specified existing workbook
BookName
The target workbook. Must be source (uneditable), new or existing (editable), otherwise blank.
Sheet
  • <new>: A new worksheet.
  • <source>: The source data worksheet
  • <existing>: A specified existing worksheet
SheetName
Name of the target sheet.
Optional Report Tables Specifies what is output to report worksheet
Notes
Notes table
Input Data
Table for input data
Masked Data
Table for masked data
Missing Data
Table for missing data

Plots

Histograms Outputs a histogram to the result sheet.

When this box is selected, the branch is expanded. In this branch, the Automatic Binning check box is selected by default. Bin Size, Number of Bins, Begin, End are shown automatically when Automatic Binning is clear.

Note: If the input data are in multiple columns, Origin will display non-editable limit values for each column. If the Automatic Binning check box is cleared, these items will be editable with an initial value.

Auto Binning.png

After you edit the value of Bin Size, Beginning and/or End, the Number of Bins will be computed automatically.

Number of Bins and Bin Size can be switched by Step By.

Number of Bins = (Beginning-End) / Bin Size

Box Charts Outputs a box chart to the report sheet. If the input data has a group column, the box chart is grouped accordingly. If the group column is Set as Categorical, the box chart will be plotted according to categorical order customized through (Column Properties) Categories tab from grouping range.