绿泥石标准差

create table t (row int);
insert into t values (1),(2),(3);
SELECT AVG((t.row - sub.a) * (t.row - sub.a)) as var from t, 
    (SELECT AVG(row) AS a FROM t) AS sub;
0.666666666666667

http://www.sqlite.org/lang_aggfunc.html

extension-functions.c this page

import sqlite3
import math

class StdevFunc:
    def __init__(self):
        self.M = 0.0
        self.S = 0.0
        self.k = 1

    def step(self, value):
        if value is None:
            return
        tM = self.M
        self.M += (value - tM) / self.k
        self.S += (value - tM) * (value - self.M)
        self.k += 1

    def finalize(self):
        if self.k < 3:
            return None
        return math.sqrt(self.S / (self.k-2))

with sqlite3.connect(':memory:') as con:

    con.create_aggregate("stdev", 1, StdevFunc)

    cur = con.cursor()

    cur.execute("create table test(i)")
    cur.executemany("insert into test(i) values (?)", [(1,), (2,), (3,), (4,), (5,)])
    cur.execute("insert into test(i) values (null)")
    cur.execute("select avg(i) from test")
    print("avg: %f" % cur.fetchone()[0])
    cur.execute("select stdev(i) from test")
    print("stdev: %f" % cur.fetchone()[0])

avg: 3.000000
stdev: 1.581139

http://sqlfiddle.com/#!2/ad42f3/3/0

Welford's method extension-functions.c

$db->sqliteCreateAggregate('stdev',
    function (&$context, $row, $data) // step callback
    {
        if (isset($context) !== true) // $context is null at first
        {
            $context = array
            (
                'k' => 0,
                'm' => 0,
                's' => 0,
            );
        }

        if (isset($data) === true) // the standard is non-NULL values only
        {
            $context['s'] += ($data - $context['m']) * ($data - ($context['m'] += ($data - $context['m']) / ++$context['k']));
        }

        return $context;
    },
    function (&$context, $row) // fini callback
    {
        if ($context['k'] > 0) // return NULL if no non-NULL values exist
        {
            return sqrt($context['s'] / $context['k']);
        }

        return null;
    },
1);

$db

select ((sum(value)*sum(value) - sum(value * value))/((count(*)-1)*(count(*)))) 
from the_table ;

class StdevFunc:
    """
    For use as an aggregate function in SQLite
    """
    def __init__(self):
        self.M = 0.0
        self.S = 0.0
        self.k = 0

    def step(self, value):
        try:
            # automatically convert text to float, like the rest of SQLite
            val = float(value) # if fails, skips this iteration, which also ignores nulls
            tM = self.M
            self.k += 1
            self.M += ((val - tM) / self.k)
            self.S += ((val - tM) * (val - self.M))
        except:
            pass

    def finalize(self):
        if self.k <= 1: # avoid division by zero
            return none
        else:
            return math.sqrt(self.S / (self.k-1))

#!/usr/bin/python
# -*- coding: utf-8 -*-
#Values produced by this script can be verified by follwing the steps
#found at https://support.microsoft.com/en-us/kb/213930 to Verify
#by chosing a non memory based database.
import sqlite3
import math
import random
import os
import sys
import traceback
import random

class StdevFunc:
    def __init__(self):
        self.M = 0.0    #Mean
        self.V = 0.0    #Used to Calculate Variance
        self.S = 0.0    #Standard Deviation
        self.k = 1      #Population or Small 

    def step(self, value):
        try:
            if value is None:
                return None

            tM = self.M
            self.M += (value - tM) / self.k
            self.V += (value - tM) * (value - self.M)
            self.k += 1
        except Exception as EXStep:
            pass
            return None    

     def finalize(self):
        try:
            if ((self.k - 1) < 3):
                return None

            #Now with our range Calculated, and Multiplied finish the Variance Calculation
            self.V = (self.V / (self.k-2))

            #Standard Deviation is the Square Root of Variance
            self.S = math.sqrt(self.V)

            return self.S
        except Exception as EXFinal:
            pass
            return None 

def Histogram(Population):
    try:
        BinCount = 6 
        More = 0

        #a = 1          #For testing Trapping
        #b = 0          #and Trace Back
        #c = (a / b)    #with Detailed Info

        #If you want to store the Database
        #uncDatabase = os.path.join(os.getcwd(),"BellCurve.db3")
        #con = sqlite3.connect(uncDatabase)

        #If you want the database in Memory
        con = sqlite3.connect(':memory:')    

        #row_factory allows accessing fields by Row and Col Name
        con.row_factory = sqlite3.Row

        #Add our Non Persistent, Runtime Standard Deviation Function to the Database
        con.create_aggregate("Stdev", 1, StdevFunc)

        #Lets Grab a Cursor
        cur = con.cursor()

        #Lets Initialize some tables, so each run with be clear of previous run
        cur.executescript('drop table if exists MyData;') #executescript requires ; at the end of the string
        cur.execute("create table IF NOT EXISTS MyData('ID' INTEGER PRIMARY KEY   AUTOINCREMENT, 'Val' FLOAT)")
        cur.executescript('drop table if exists Bins;')   #executescript requires ; at the end of the string
        cur.execute("create table IF NOT EXISTS Bins('ID' INTEGER PRIMARY KEY   AUTOINCREMENT, 'Bin' UNSIGNED INTEGER, 'Val' FLOAT, 'Frequency' UNSIGNED BIG INT)")

        #Lets generate some random data, and insert in to the Database
        for n in range(0,(Population)):
            sql = "insert into MyData(Val) values ({0})".format(random.uniform(-1,1))
            #If Whole Number Integer greater that value of 2, Range Greater that 1.5
            #sql = "insert into MyData(Val) values ({0})".format(random.randint(-1,1))
            cur.execute(sql)
            pass

        #Now let’s calculate some built in Aggregates, that SQLite comes with
        cur.execute("select Avg(Val) from MyData")
        Average = cur.fetchone()[0]
        cur.execute("select Max(Val) from MyData")
        Max = cur.fetchone()[0]
        cur.execute("select Min(Val) from MyData")
        Min = cur.fetchone()[0]
        cur.execute("select Count(Val) from MyData")
        Records = cur.fetchone()[0]

        #Now let’s get Standard Deviation using our function that we added
        cur.execute("select Stdev(Val) from MyData")
        Stdev = cur.fetchone()[0]

        #And Calculate Range
        Range = float(abs(float(Max)-float(Min)))

        if (Stdev == None):
            print("================================   Data Error ===============================")
            print("                 Insufficient Population Size, Or Bad Data.")   
            print("*****************************************************************************")
        elif (abs(Max-Min) == 0):
            print("================================   Data Error ===============================")
            print(" The entire Population Contains Identical values, Distribution Incalculable.")
            print("******************************************************************************")            
        else:  
            Bin = []        #Holds the Bin Values
            Frequency = []  #Holds the Bin Frequency for each Bin

            #Establish the 1st Bin, which is based on (Standard Deviation * 3) being subtracted from the Mean
            Bin.append(float((Average - ((3 * Stdev)))))
            Frequency.append(0)

            #Establish the remaining Bins, which is basically adding 1 Standard Deviation
            #for each interation, -3, -2, -1, 1, 2, 3             
            for b in range(0,(BinCount) + 1):
                Bin.append((float(Bin[(b)]) + Stdev))
                Frequency.append(0)

            for b in range(0,(BinCount) + 1):
                #Lets exploit the Database and have it do the hard work calculating distribution
                #of all the Bins, with SQL's between operator, but making it left inclusive, right exclusive.
                sqlBinFreq = "select count(*) as Frequency from MyData where val between {0} and {1} and Val < {2}". \
                             format(float((Bin[b])), float(Bin[(b + 1)]), float(Bin[(b + 1)]))

                #If the Database Reports Values that fall between the Current Bin, Store the Frequency to a Bins Table. 
                for rowBinFreq in cur.execute(sqlBinFreq):
                    Frequency[(b + 1)] = rowBinFreq['Frequency']
                    sqlBinFreqInsert = "insert into Bins (Bin, Val, Frequency) values ({0}, {1}, {2})". \
                                   format(b, float(Bin[b]), Frequency[(b)])
                    cur.execute(sqlBinFreqInsert)

                #Allthough this Demo is not likley produce values that
                #fall outside of Standard Distribution
               #if this demo was to Calculate with real data, we want to know
               #how many non-Standard data points we have. 
               More = (More + Frequency[b])

            More = abs((Records - More))

            #Add the More value
            sqlBinFreqInsert = "insert into Bins (Bin, Val, Frequency) values ({0}, {1}, {2})". \
                            format((BinCount + 1), float(0), More)
            cur.execute(sqlBinFreqInsert)

            #Now Report the Analysis
            print("================================ The Population ==============================")
            print("             {0} {1} {2} {3} {4} {5}". \
              format("Size".rjust(10, ' '), \
                     "Max".rjust(10, ' '), \
                     "Min".rjust(10, ' '), \
                     "Mean".rjust(10, ' '), \
                     "Range".rjust(10, ' '), \
                     "Stdev".rjust(10, ' ')))
            print("Aggregates:  {0:10d} {1:10.4f} {2:10.4f} {3:10.4f} {4:10.4f} {5:10.4f}". \
              format(Population, Max, Min, Average, Range, Stdev))
             print("================================= The Bell Curve =============================")  

            LabelString = "{0} {1}  {2}  {3}". \
                      format("Bin".ljust(8, ' '), \
                             "Ranges".rjust(8, ' '), \
                             "Frequency".rjust(8, ' '), \
                             "Histogram".rjust(6, ' '))

            print(LabelString)
            print("------------------------------------------------------------------------------")

            #Let's Paint a Histogram
            sqlChart = "select * from Bins order by Bin asc"
            for rowChart in cur.execute(sqlChart):
                if (rowChart['Bin'] == 7):
                    #Bin 7 is not really a bin, but where we place the values that did not fit into the
                    #Normal Distribution. This script was tested against Excel's Bell Curve Example
                    #https://support.microsoft.com/en-us/kb/213930
                    #and produces the same results. Feel free to test it.
                    BinName = "More"
                    ChartString = "{0:<6} {1:<10} {2:10.0f}". \
                            format(BinName, \
                                    "", \
                                    More)
                else:
                    #Theses are the actual bins where values fall within the distribution.
                    BinName = (rowChart['Bin'] + 1)
                    #Scale the Chart
                    fPercent = ((float(rowChart['Frequency']) / float(Records) * 100))
                    iPrecent = int(math.ceil(fPercent))

                    ChartString = "{0:<6} {1:10.4f} {2:10.0f}  {3}". \
                              format(BinName, \
                                     rowChart['Val'], \
                                     rowChart['Frequency'], \
                                     "".rjust(iPrecent, '#'))
                print(ChartString)

            print("******************************************************************************")

            #Commit to Database
            con.commit()

            #Clean Up
            cur.close()
            con.close()

    except Exception as EXBellCurve:
        pass
        TraceInfo = traceback.format_exc()       
        raise Exception(TraceInfo)