Question

The following is a sample of 25 measurements:

(7, 6, 6, 11, 8, 9, 11, 9, 10, 8, 7, 7, 5, 9, 10, 7, 7, 7, 7, 9, 12, 10, 10, 8, 6)

a.Compute x, s2,, and sfor this sample.

b.Count the number of measurements in the intervals xbar ± s, xbar ± 2s, xbar ± 3s. Express each count as a percentage of the total number of measurements.

c.Compare the percentages found in part bto the percentages given by the Empirical Rule and Chebyshev’s Rule.

d.Calculate the range and use it to obtain a rough approximation for s. Does the result compare favorably with the actualvaluefor sfound in part a?

Short answer

a. xbar = 8.24, s² = 3.356, s = 1.83

b. 19, 76%

24, 96%

25, 100%

c. Follows the Empirical Rule

d. Range =6, s = 1.5

Step-by-step solution

Computing x bar, s2, s

x	$(x-\overline{x})$	$(x-\overline{x}{)}^2$
7	-1.24	1.5376
6	-2.24	5.0176
6	-2.24	5.0176
11	2.76	7.6176
8	-0.24	0.0576
9	0.76	0.5776
11	2.76	7.6176
9	0.76	0.5776
10	1.76	3.0976
8	-0.24	0.0576
7	-1.24	1.5376
7	-1.24	1.5376
5	-1.066	1.136
9	0.76	0.5776
10	1.76	3.0976
7	-1.24	1.5376
7	-1.24	1.5376
7	-1.24	1.5376
7	-1.24	1.5376
9	0.76	0.5776
12	3.76	14.1376
10	1.76	3.0976
10	1.76	3.0976
8	-0.24	0.0576
6	-2.24	5.0176
Sum=206	0	80.56

localid="1668425769647" $\begin{array}{rcl}\mathbf{Mean}\mathbf{\left(}\overline{\mathbf{x}}\mathbf{\right)}& \mathbf{=}& \frac{\mathbf{\sum }\mathbf{x}}{\mathbf{n}}\\ & \mathbf{=}& \frac{\mathbf{206}}{\mathbf{25}}\\ & \mathbf{=}& \mathbf{8}\mathbf{.}\mathbf{24}\\ & & \\ \mathbf{Variance}& \mathbf{=}& \frac{\mathbf{\sum }\mathbf{(}\mathbf{x}\mathbf{-}\overline{\mathbf{x}}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{n}\mathbf{-}\mathbf{1}}\\ & \mathbf{=}& \frac{\mathbf{80}\mathbf{.}\mathbf{56}}{\mathbf{25}\mathbf{-}\mathbf{1}}\\ & \mathbf{=}& \frac{\mathbf{80}\mathbf{.}\mathbf{56}}{\mathbf{24}}\\ & \mathbf{=}& \mathbf{3}\mathbf{.}\mathbf{356}\\ & & \\ \mathbf{StandardDeviation}& \mathbf{=}& \sqrt{\mathbf{Variance}}\\ & \mathbf{=}& \sqrt{\mathbf{3}\mathbf{.}\mathbf{356}}\\ & \mathbf{=}& \mathbf{1}\mathbf{.}\mathbf{83}\\ & & \end{array}$

Therefore, xbar = 8.24, s² = 3.356, s = 1.83.

Counting the number of measurements in various ranges

Minimum = 6

Maximum = 12

$\mathbf{(}\overline{\mathbf{x}}\mathbf{-}\mathit{s}\mathbf{,}\overline{\mathbf{x}}\mathbf{+}\mathit{s}\mathbf{)}$

$\begin{array}{rcl}\overline{\mathbf{x}}& \mathbf{=}& \mathbf{8}\mathbf{.}\mathbf{24}\\ \mathbf{s}& \mathbf{=}& \mathbf{1}\mathbf{.}\mathbf{83}\\ \mathbf{Lowerrange}& \mathbf{=}& \overline{\mathbf{x}}\mathbf{-}\mathbf{s}\\ & \mathbf{=}& \mathbf{8}\mathbf{.}\mathbf{24}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{83}\\ & \mathbf{=}& \mathbf{6}\mathbf{.}\mathbf{41}\\ \mathbf{Upperrange}& \mathbf{=}& \overline{\mathbf{x}}\mathbf{+}\mathbf{s}\\ & \mathbf{=}& \mathbf{8}\mathbf{.}\mathbf{24}\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{83}\\ & \mathbf{=}& \mathbf{10}\mathbf{.}\mathbf{07}\\ & & \end{array}$

Now that we have our range, we will count the measurements that fall within this range to get the number of measurements in this range.

19 measurements fall in the range.

To get the percentage, we will have to divide the number of measurements in the range by the total measurements.

$\begin{array}{rcl}\frac{\mathbf{24}}{\mathbf{25}}& \mathbf{=}& \mathbf{0}\mathbf{.}\mathbf{76}\\ \mathbf{0}\mathbf{.}\mathbf{76}\mathbf{\times }\mathbf{100}& \mathbf{=}& \mathbf{76}\mathbf{\%}\\ & & \end{array}$

Therefore, 76% of our measurements fall in this range.

$\mathbf{(}\overline{\mathbf{x}}\mathbf{-}\mathbf{2}\mathit{s}\mathbf{,}\overline{\mathbf{x}}\mathbf{+}\mathbf{2}\mathit{s}\mathbf{)}$

localid="1668425792067" $\begin{array}{rcl}\overline{\mathbf{x}}& \mathbf{=}& \mathbf{8}\mathbf{.}\mathbf{24}\\ \mathbf{s}& \mathbf{=}& \mathbf{1}\mathbf{.}\mathbf{83}\\ \mathbf{Lowerrange}& \mathbf{=}& \overline{\mathbf{x}}\mathbf{-}\mathbf{2}\mathbf{s}\\ & \mathbf{=}& \mathbf{8}\mathbf{.}\mathbf{24}\mathbf{-}\mathbf{2}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{83}\mathbf{)}\\ & \mathbf{=}& \mathbf{4}\mathbf{.}\mathbf{58}\\ \mathbf{Upperrange}& \mathbf{=}& \overline{\mathbf{x}}\mathbf{+}\mathbf{2}\mathbf{s}\\ & \mathbf{=}& \mathbf{8}\mathbf{.}\mathbf{24}\mathbf{+}\mathbf{2}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{83}\mathbf{)}\\ & \mathbf{=}& \mathbf{11}\mathbf{.}\mathbf{9}\\ & & \end{array}$

To get the number of measurements in the range of 4.58 to 11.9, we will count the measurements within this range.

Twenty-four measurements fall in this range.

To get the percentage, we will have to divide the number of measurements in the range by the total measurements.

$\begin{array}{rcl}\frac{\mathbf{24}}{\mathbf{25}}& \mathbf{=}& \mathbf{0}\mathbf{.}\mathbf{96}\\ \mathbf{0}\mathbf{.}\mathbf{96}\mathbf{\times }\mathbf{100}& \mathbf{=}& \mathbf{96}\mathbf{\%}\\ & & \end{array}$

Therefore, 96% of our measurements fall in the 2 standard deviation range.

$\mathbf{(}\overline{\mathbf{x}}\mathbf{-}\mathbf{3}\mathit{s}\mathbf{,}\overline{\mathbf{x}}\mathbf{+}\mathbf{3}\mathit{s}\mathbf{)}$

localid="1668425810731" $\begin{array}{rcl}\overline{\mathbf{x}}& \mathbf{=}& \mathbf{8}\mathbf{.}\mathbf{24}\\ \mathbf{s}& \mathbf{=}& \mathbf{1}\mathbf{.}\mathbf{83}\\ \mathbf{Lowerrange}& \mathbf{=}& \overline{\mathbf{x}}\mathbf{-}\mathbf{3}\mathbf{s}\\ & \mathbf{=}& \mathbf{8}\mathbf{.}\mathbf{24}\mathbf{-}\mathbf{3}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{83}\mathbf{)}\\ & \mathbf{=}& \mathbf{2}\mathbf{.}\mathbf{75}\\ \mathbf{Upperrange}& \mathbf{=}& \overline{\mathbf{x}}\mathbf{+}\mathbf{2}\mathbf{s}\\ & \mathbf{=}& \mathbf{8}\mathbf{.}\mathbf{24}\mathbf{+}\mathbf{3}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{83}\mathbf{)}\\ & \mathbf{=}& \mathbf{13}\mathbf{.}\mathbf{73}\\ & & \end{array}$

Now that we have our range, we will count the measurements that fall within this range to get the number of measurements in this range.

All 25 measurements fall in this range.

To get the percentage, we will have to divide the number of measurements in the range by the total measurements.

$\begin{array}{rcl}\frac{\mathbf{25}}{\mathbf{25}}& \mathbf{=}& \mathbf{1}\\ \mathbf{1}\mathbf{\times }\mathbf{100}& \mathbf{=}& \mathbf{100}\mathbf{\%}\\ & & \end{array}$

Therefore, 100% of our measurements fall in the 3 standard deviation range.

Comparing the percentages found in part b to the percentages given by the Empirical Rule and Chebyshev’s Rule

According to the Chebyshev rule, very few or none of the measurements fall in the $\mathbf{(}\overline{\mathbf{x}}\mathbf{-}\mathit{s}\mathbf{,}\overline{\mathbf{x}}\mathbf{+}\mathit{s}\mathbf{)}$range and based on the Empirical rule, almost 68% of the measurements fall in this range.

In the 2 standard deviation range, ¾ or 75% of measurements lie here as per the Chebyshev rule and almost 95% according to the Empirical rule.

In the 3 standard deviation range 8/9, or almost 89% measurements are here, and according to the Empirical rule, 99.6% are here.

Now that we know the rules and have the values in part a, we can clearly say that this distribution is in mound shape and conforms to the Empirical Rule.

Computing Range and using it to obtain s and comparing it to the actual s

Range = Max – Min

= 12 – 6

= 6

$\begin{array}{rcl}\mathit{s}& \mathbf{=}& \frac{\mathbf{R}\mathbf{a}\mathbf{n}\mathbf{g}\mathbf{e}}{\mathbf{4}}\\ & \mathbf{=}& \frac{\mathbf{6}}{\mathbf{4}}\\ & \mathbf{=}& \mathbf{1}\mathbf{.}\mathbf{5}\\ & & \end{array}$

Using the range, the standard deviation is approximately 1.5.

The actual standard deviation is 1.83.

The approximate standard deviation is lower than the actual one.