Because an exact number for standard deviation could only be obtained by taking an infinite number of measurements, the below formula calculates the estimated standard deviation.

Let’s take a set of measurements from a temperature transmitter with a range of 0-100 deg, Celsius.

IDEAL VALUES |
AS FOUND VALUES |

0 | 1 |

10 | 12 |

20 | 21 |

30 | 28 |

40 | 42 |

50 | 51 |

60 | 62 |

70 | 68 |

80 | 85 |

90 | 91 |

100 | 102 |

The first step for calculating for estimated standard deviation will be to find our average deviation or mean.

For our example the deviations are:

1,2,1,2,2,1,2,2,5,1,2

The average deviation would be 1.909 degrees Celsius.

The next step is to determine how far off each measurement is from the average deviation.

.9,.1,.9,.1,.1,.9,.1,.1,3.1,.9,.1

Then we take the square of each of these and sum.

.81+.01+.81+.01+.01+.81+.01+.01+9.61+.81+.01=12.91

Then we divide by the number of measurements minus 1, which is 10, to arrive at 1.29

The square root of 1.29 is 1.136 and is our estimated standard deviation.

The Estimated standard deviation, based upon the 11 checks made, is 1.136 degrees Celsius for the transmitter.

Estimated Standard Deviation is expressed in same units as data.

**References**:

*A Beginner’s Guide to Uncertainty of Measurement *by Stephanie Bell.