====== ND Summary Statistics ======

The Normal Database (ND) feature in Sift includes a library of summary statistics that the user can choose from. 

Here we list all available summary statistics that can be computed against the ND library. 

If there are statistics you wish to calculate they are not included here, email info@has-motion.ca and let us know!



===== Measures of Central Tendencies =====



==== Mean ====
**Definition:** The sum of all values divided by the total number of values.\\

**Equation:** {{:sift:build_nds:meaneq.png?125}}
 
Xi = each score in the dataset\\
n= sample size\\

//**Note:** this measure is not a weighted mean or mean of means. If measuring mean of a workspace or library summary level in the Normal Database, the mean is measured across all traces from all trials and workspaces (if Library level summary) in the dataset. //\\
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==== Median ====

**Definition:** The value lying at the midpoint of the sorted dataset.\\

**Equation:**\\
  * //if odd// {{:sift:build_nds:medeqodd.png?125}}
  * //if even// {{:sift:build_nds:medeqeven.png?150}}
  * n is the data set/sample size
  * X is the ordered list of values in the data set

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==== Mode ====
**Definition:** The value that appears most often in a data set.\\

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==== Trimean ====
**Definition:** A weighted average of the median and quartiles of the dataset.\\

**Equation:** (quartile1 + median*2 + quartile3)/4


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==== Geometric Mean ====
**Definition:** An average of a data set using the product of the values rather than the sum. The nth root of the product of n positive numbers.\\

**Equation:** {{:sift:build_nds:gmean.png?240}}

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==== Trimmed Mean ====
**Definition:** Calculates the mean after removing a percentage of the smallest and largest values in the dataset. In this case, we are using a trimmed mean of 10%\\

**Equation:** {{:sift:build_nds:meaneq.png?125}}
 
Xi = each score in the trimmed data set in ascending order\\
n= sample size of the trimmed data set\\

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===== Measures of Dispersion =====

==== Standard Deviation ====
**Definition:** The square root of the variance of the dataset relative to it's mean.\\

**Equation:** {{:sift:build_nds:stdeveq.png?125}}
 
Xi = each score in the data set\\
u = mean of the data set\\
n= sample size of the data set\\

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==== Variance ====
**Definition:** The mean of square differences between each data point and the mean of the data set.\\

**Equation:** {{:sift:build_nds:varianceeq.png?125}}
 
Xi = each score in the data set\\
u = mean of the data set\\
n= sample size of the data set\\

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==== Range ====
**Definition:** The size of the narrowest interval that contains all data points in the dataset. Found as the difference between the largest and smallest values in the data set.\\

**Equation:** {{:sift:build_nds:rangeeq.png?200}}
 
X = the data set\\

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==== Quartiles ====
**Definition:** The division of a data set (in ascending order) into four equal portions, each containing 25% of the data points.\\

**Equation:** {{:sift:build_nds:quartileeqn.png?200}}
 
N = the number of points in the data set\\

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==== Interquartile Range ====
**Definition:** The range of the middle 50% of the data set. Found by finding the difference between the third quartile (75th percentile) and first quartile (25th percentile). \\

**Equation:** {{:sift:build_nds:IQReq.png?125}}
 

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==== Percentiles ====
**Definition:** The value of a sorted data set where x% of the data is below that value.\\

**Equation:** {{:sift:build_nds:percentileseq.png?125}}\\
{{:sift:build_nds:percentileseq2.png?100}}\\
x = percentile of interest\\
Pi = index of xth percentile in the data set\\
n = total number of points in the data set\\
X = sorted data set\\
Px = the data value at index Pi\\

//Note: in this implementation, we provide the 5th and 95th percentiles.//

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==== Mean Absolute Deviation ====
**Definition:** The measure of variability representing the average distance of each data point and the mean of the data set.\\

**Equation:** {{:sift:build_nds:madeq.png?125}}\\
N = the number of points in the data set\\
u = the **mean** of the data set\\
Xi = each data point in the data set\\

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==== Median Absolute Deviation ====
**Definition:** The measure of variability representing the average distance of each data point and the median of the data set.\\

**Equation:** {{:sift:build_nds:madeq.png?125}}\\
N = the number of points in the data set\\
u = the **median** of the data set\\
Xi = each data point in the data set\\