Question

For the numbers 116.0,97.9,114.2,106.8and 108.3, find the mean, standard deviation, standard uncertainty ( = standard deviation of the mean), range, and 90% confidence interval for the mean. Using the Grubbs test, decide whether the number 97.9should be discarded.

Short answer

If$G_{calculated}>G_{uble}$, then the questionable point should be discarded.

If$G_{ealculated}<G_{table}$ , then the questionable point should be retained.

Step-by-step solution

Definition of standard deviation and mean.

• Arithmetic mean: By dividing the amount of measured values by the number of measurements n , the arithmetic mean $\overline{x}$ is determined. The average is often referred to as the "mean."

Mean:$\overline{x}=\frac{{\displaystyle \sum _i}{x}_i}{n}$

• Standard deviation: The standard deviation is a measurement of how tightly the data cluster around the mean.

The standard deviation (s) is given by the formula:

$s=\sqrt{\frac{{\displaystyle \sum _i}{\left(x_i-\overline{x}\right)}^2}{n-1}}$

Standard deviation of the mean of sets ofvalues is:

${\sigma }_n=\frac{\theta }{\sqrt{n}}$

• Range: The difference between the highest value and the lowest value in a set of data gives the range.
• Confidence Intervals: The confidence interval is given by the equation:

Confidence interval$$\begin{gathered} =\overline{x}\pm \frac{ts}{\sqrt{n}} \\ =\overline{x}\pm t{u}_x \end{gathered}$$

(since standard uncertainty$\left(u_x\right)=s/\sqrt{n}$ )

Where, $\overline{x}$is mean, s is standard deviation, t is Student's, n is number of measurements, $u_x$is standard uncertainty.

Find the mean standard deviation, standard uncertainty, range and 90% confidence interval for the mean.

It is necessary to determine the mean, standard deviation, standard uncertainty, range, and 90% confidence interval for the mean.

The Grubbs test will also determine if the number 97.9 should be rejected or kept.

Grubbs Test for an Outlier:

An outlier is a point that is remote from the rest of the data. Grubbs test determines whether the datum should be considered or rejected from the complete set of points.

The Grubbs statistic G, is given as:

$G_{calculated}=\frac{\left|kueationablevalue-\overline{x}\right|}{s}$

Where,

$\overline{x}$is the mean of entire data set.

s is the standard deviation of entire data set.

If$G_{calculated}>G_{uble}$ , then the questionable point should be discarded.

If$G_{ealculamed}<G_{table}$ , then the questionable point should be retained.