3.6 Confidence intervals  3. Probability Distribution Functions  3.4 Poisson distribution

3.5 Student's t distribution

Suppose that a set of observations has mean $\overline{x}$. To test the hypothesis that this sample came from a population with mean $\mu$ obeying a Gaussian distribution, we might try the test statistic

$$z = {{\overline{x}-\mu}\over{\sigma/\sqrt{n}}} ={{\overline{x}-\mu}\over{\sigma_\mu}}$$ (3.8)

 
where $\sigma$ is the true standard deviation in x and $\sigma_\mu=\sigma/sqrt{n}$ is the standard deviation in the mean. However, usually the true standard deviation $\sigma$for the population from which the sample was collected is unknown. An estimator for $\sigma$, calculated from the observations, is

$$s = \Bigl[{{1}\over{n-1}}\sum_i(x_i-\overline{x})^2\Bigr]^{1/2}.$$ (3.9)

 

A candidate test statistic is thus

$$t = {{\overline{x}-\mu}\over{s/\sqrt{n}}} = {{\overline{x}-\mu}\over{s_\mu}}$$ (3.10)

 
where $s_\mu=s/\sqrt{n}$.

Although z would obey a Gaussian distribution if the individual measurements entering $\overline{x}$ do, t will not be Gaussian distributed. Instead, the distribution in t is determined by the ratio of the Gaussian distribution to the square root of the chisquare distribution which characterizes deviations of the sample estimate of the standard deviation from the true standard deviation:^3.1

$$t = {{\overline{x}-\mu}\over{\sigma/\sqrt{n}}}\Bigl[{{\sigma^2}\over{s^2}}\Bigl]^{1/2}$$ (3.11)

 

where $s^2/\sigma^2$ is distributed as $\chi^2(n-1)/(n-1)$.

The distribution in t can be derived from this ratio (as shown, e.g., in Brownlee 1965):

$$\Phi_t(t,\nu) = {{\Gamma\bigl({{\nu+1}\over{2}}\bigr)}\over{......}\bigr)}}\bigl(1+{{t^2}\over{\nu}}\bigr)^{-{{\nu+1}\over{2}}}$$ (3.12)

 

where $\nu$ is the number of degrees of freedom, which will be (n-1) in the case where $\overline{x}$ is the average of n independent measurements. The variance of the t distribution is given by

$$V_{tt} = {{\nu}\over{\nu-2}} \ ,$$ (3.13)

 

and for large numbers of degrees of freedom the t distribution approaches a Gaussian distribution with this variance.

The t statistic supports a test of the hypothesis that the mean is $\mu$ without requiring that the true standard deviation $\sigma$ be known. Only the sample standard deviation is needed. The change from a Gaussian distribution function is most significant for small numbers of degrees of freedom. Figure 2.1 showed the cumulative form of this distribution function for various degrees of freedom and compared its shape to that of the Gaussian distribution.
 

Example 3.1: Two instruments measuring the concentration of droplet concentration in clouds collect measurements that, by linear regression, are related by a slope parameter b and a standard deviation s_b. We want to test if these results are consistent with a slope of 1, as would be expected if the two instruments were calibrated and operating consistently. The regression fit has N-2 degrees of freedom if N measurements are used in the comparison, so the appropriate test statistic is

$$t={{\overline{b}-1}\over{\sigma_b}}$$ (3.14)

 

with N-2 degrees of freedom. This can be used to test the probability that a value as large as t will be obtained if the true mean is 1.

Example 3.2: Test if two experimental results are in conflict or are consistent with expected deviations in repeated experiments. Suppose that two different experimenters obtain, respectively, $R_1\pm s_1$ and $R_2\pm s_2$. To test for consistency of these results, use the difference (R₂-R₁). The standard deviation in this difference can be estimated from

$$s = \sqrt{s_1^2+s_2^2} .$$ (3.15)

 

Then

$$t={{R_2-R_1}\over{s}}$$ (3.16)

 

is distributed according to the t distribution with (n₁+n₂-1) degrees of freedom, where n₁ and n₂ are the respective degrees of freedom in the two experiments. A common mistake is to assume that the degrees of freedom should be 1 for this case because there are two measurements and one difference. Recall that the reason for introducing the t distribution was to account for lack of knowledge of the true standard deviation in the phenomenon being observed. If there is a large set of measurements, this standard deviation is known much better than if there are only a few, and this improved knowledge must be reflected in the number of degrees of freedom in the comparison.

   
 3.6 Confidence intervals  3. Probability Distribution Functions  3.4 Poisson distribution