Study text: "Essential Mathematics and Statistics for Science", 2nd Edition,

G Currell & A A Dowman, Wiley-Blackwell, 2009

G Currell & A A Dowman, Wiley-Blackwell, 2009

- When dealing with experimental data, we calclulate the
**sample**standard deviation (function STDEV in Excel) rather than the**population**standard deviation (function STDEVP in Excel) because:-- the sample standard deviation gives the larger value and makes a better allowance for actual errors
- the sample standard deviation is an estimate of the true population standard deviation of the measurement being made
- the sample standard deviation was the easier value to calculate before computers became widely available

- In an experiment, four repeated (replicate) measurements of the same gas pressure were made to obtain an estimate of the
**uncertainty**in the pressure measurement. The values obtained, 58, 59, 56 and 63 mbar, were entered into an Excel worksheet between cells B4 to B7.

The Excel function STDEV(B4:B7) in a separate cell calculated the**sample standard deviation**of the**4**data values, giving a value of 2.94 mbar.

If**400**replicate measurements were then made of the same pressure, we would expect that the sample standard deviation for these 400 measurements:-- to remain equal to about 3 mbar
- to reduce to about 0.3 mbar
- to reduce to about 0.03 mbar

- If the true value for pressure were 60 mbar, and the standard deviation of experimental measurements were 3 mbar, we expect that the fraction of
**experimental values**falling within**one standard deviation**of the true value (i.e. between 57 and 63) will be approximately:-- half
- two-thirds
- 95%

- Four,
*n*, experimental measurements of gas pressure, 58, 59, 56 and 63 mbar, give a mean value,*p*= 59.0 mbar and a sample standard deviation,*s*= 2.94 mbar. The best estimate for the unknown true value is the calculated mean value, 59.0 mbar. A frequently used measure of the uncertainty, or possible error, in the estimated true value is the**standard uncertainty**(or**standard error**),*u*, which is given by - Four,
*n*, experimental measurements of gas pressure, 58, 59, 56 and 63 mbar, give a mean value,*p*= 59.0 mbar and a sample standard deviation,*s*= 2.94 mbar. The**‘95% Confidence Interval’**specifies the range of values within which it is possible to be 95% confident that the unknown true value will lie.

On the basis of the experimental values, it is possible to state, with 95% confidence, that the unknown true value of the gas pressure will be within the range:

(*t*-values for 2, 3, 4 and 16 degrees of freedom respectively are 4.30, 3.18, 2.78 and 2.12)- 59.0 ± 4.30×2.94/√4, i.e. between 52.7 and 65.3
- 59.0 ± 3.18×2.94/√4, i.e. between 54.3 and 63.7
- 59.0 ± 2.78×2.94/√4, i.e. between 54.9 and 63.1
- 59.0 ± 2.0×2.94/√4, i.e. between 56.1 and 61.9