Question

the number of incorrect answers on a true - or - false mathematics proficiency test for a random sample of 20 students was recorded as follows: 3 1 5 4 1 2 4 4 5 1 3 3 2 5 4 4 5 1 2 find the: 1. range 2. mean absolute deviation 3. variance 4. standard deviation 5. quartile deviation

Explanation:

Step1: Arrange data in ascending order

1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6

Step2: Calculate the range

The range is the difference between the maximum and minimum values.
Range = 6 - 1 = 5

Step3: Calculate the mean ($\bar{x}$)

$\bar{x}=\frac{1\times5 + 2\times3+3\times3 + 4\times5+5\times3+6\times1}{20}=\frac{5 + 6+9 + 20+15+6}{20}=\frac{61}{20}=3.05$

Step4: Calculate the absolute - deviations

$x_i$	$	x_i-\bar{x}	$
1	$	1 - 3.05	=2.05$
1	$	1 - 3.05	=2.05$
1	$	1 - 3.05	=2.05$
1	$	1 - 3.05	=2.05$
2	$	2 - 3.05	=1.05$
2	$	2 - 3.05	=1.05$
2	$	2 - 3.05	=1.05$
3	$	3 - 3.05	=0.05$
3	$	3 - 3.05	=0.05$
3	$	3 - 3.05	=0.05$
4	$	4 - 3.05	=0.95$
4	$	4 - 3.05	=0.95$
4	$	4 - 3.05	=0.95$
4	$	4 - 3.05	=0.95$
4	$	4 - 3.05	=0.95$
5	$	5 - 3.05	=1.95$
5	$	5 - 3.05	=1.95$
5	$	5 - 3.05	=1.95$
6	$	6 - 3.05	=2.95$

Step5: Calculate the mean absolute deviation (MAD)

$MAD=\frac{\sum_{i = 1}^{n}|x_i-\bar{x}|}{n}=\frac{2.05\times5+1.05\times3 + 0.05\times3+0.95\times5+1.95\times3+2.95\times1}{20}=\frac{10.25+3.15 + 0.15+4.75+5.85+2.95}{20}=\frac{27.1}{20}=1.355$

Step6: Calculate the squared - deviations

$x_i$	$(x_i-\bar{x})^2$
1	$(1 - 3.05)^2=4.2025$
1	$(1 - 3.05)^2=4.2025$
1	$(1 - 3.05)^2=4.2025$
1	$(1 - 3.05)^2=4.2025$
2	$(2 - 3.05)^2=1.1025$
2	$(2 - 3.05)^2=1.1025$
2	$(2 - 3.05)^2=1.1025$
3	$(3 - 3.05)^2=0.0025$
3	$(3 - 3.05)^2=0.0025$
3	$(3 - 3.05)^2=0.0025$
4	$(4 - 3.05)^2=0.9025$
4	$(4 - 3.05)^2=0.9025$
4	$(4 - 3.05)^2=0.9025$
4	$(4 - 3.05)^2=0.9025$
4	$(4 - 3.05)^2=0.9025$
5	$(5 - 3.05)^2=3.8025$
5	$(5 - 3.05)^2=3.8025$
5	$(5 - 3.05)^2=3.8025$
6	$(6 - 3.05)^2=8.7025$

Step7: Calculate the variance ($s^2$)

$s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}=\frac{4.2025\times5+1.1025\times3+0.0025\times3 + 0.9025\times5+3.8025\times3+8.7025\times1}{19}=\frac{21.0125+3.3075+0.0075+4.5125+11.4075+8.7025}{19}=\frac{48.95}{19}\approx2.5763$

Step8: Calculate the standard deviation ($s$)

$s=\sqrt{s^2}=\sqrt{2.5763}\approx1.6051$

Step9: Calculate the quartiles

The median ($Q_2$) of the 20 - data set is the average of the 10th and 11th ordered values. So $Q_2=\frac{3 + 3}{2}=3$.
The lower half of the data is 1, 1, 1, 1, 1, 2, 2, 2, 3, 3. The median of the lower half ($Q_1$) is the 5th value, so $Q_1 = 1$.
The upper half of the data is 4, 4, 4, 4, 4, 5, 5, 5, 6. The median of the upper half ($Q_3$) is the 5th value of the upper - half, so $Q_3 = 4$.
The quartile deviation (QD) is $QD=\frac{Q_3 - Q_1}{2}=\frac{4 - 1}{2}=1.5$

Answer:

1. Range: 5
2. Mean absolute deviation: 1.355
3. Variance: approximately 2.5763
4. Standard deviation: approximately 1.6051
5. Quartile Deviation: 1.5