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 3 5 6 1 2 1 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, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 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 ($\bar{x}$)

$\bar{x}=\frac{1\times3 + 2\times3+3\times4 + 4\times4+5\times4+6\times1}{20}=\frac{3 + 6+12 + 16+20+6}{20}=\frac{63}{20}=3.15$

Step4: Calculate the absolute - deviation for each data point

$x_i$	$	x_i-\bar{x}	$
1	$	1 - 3.15	=2.15$
1	$	1 - 3.15	=2.15$
2	$	2 - 3.15	=1.15$
2	$	2 - 3.15	=1.15$
2	$	2 - 3.15	=1.15$
3	$	3 - 3.15	=0.15$
3	$	3 - 3.15	=0.15$
3	$	3 - 3.15	=0.15$
3	$	3 - 3.15	=0.15$
4	$	4 - 3.15	=0.85$
4	$	4 - 3.15	=0.85$
4	$	4 - 3.15	=0.85$
4	$	4 - 3.15	=0.85$
5	$	5 - 3.15	=1.85$
5	$	5 - 3.15	=1.85$
5	$	5 - 3.15	=1.85$
5	$	5 - 3.15	=1.85$
6	$	6 - 3.15	=2.85$

Step5: Calculate the mean absolute deviation (MAD)

$MAD=\frac{3\times2.15+3\times1.15 + 4\times0.15+4\times0.85+4\times1.85+1\times2.85}{20}$
$=\frac{6.45+3.45 + 0.6+3.4+7.4+2.85}{20}=\frac{24.15}{20}=1.2075$

Step6: Calculate the squared - deviation for each data point

$x_i$	$(x_i - \bar{x})^2$
1	$(1 - 3.15)^2=4.6225$
1	$(1 - 3.15)^2=4.6225$
2	$(2 - 3.15)^2=1.3225$
2	$(2 - 3.15)^2=1.3225$
2	$(2 - 3.15)^2=1.3225$
3	$(3 - 3.15)^2=0.0225$
3	$(3 - 3.15)^2=0.0225$
3	$(3 - 3.15)^2=0.0225$
3	$(3 - 3.15)^2=0.0225$
4	$(4 - 3.15)^2=0.7225$
4	$(4 - 3.15)^2=0.7225$
4	$(4 - 3.15)^2=0.7225$
4	$(4 - 3.15)^2=0.7225$
5	$(5 - 3.15)^2=3.4225$
5	$(5 - 3.15)^2=3.4225$
5	$(5 - 3.15)^2=3.4225$
5	$(5 - 3.15)^2=3.4225$
6	$(6 - 3.15)^2=8.1225$

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

$s^2=\frac{3\times4.6225+3\times1.3225 + 4\times0.0225+4\times0.7225+4\times3.4225+1\times8.1225}{20 - 1}$
$=\frac{13.8675+3.9675+0.09+2.89+13.69+8.1225}{19}=\frac{42.6375}{19}\approx2.2441$

Step8: Calculate the standard deviation ($s$)

$s=\sqrt{s^2}=\sqrt{2.2441}\approx1.498$

Step9: Calculate the quartiles

The median (second - quartile $Q_2$) of the 20 - data set is the average of the 10th and 11th ordered values.
$Q_2=\frac{3 + 4}{2}=3.5$
The first - quartile $Q_1$ is the median of the lower half (the first 10 values). The lower half is 1, 1, 1, 2, 2, 2, 3, 3, 3, 3. So $Q_1 = 2$
The third - quartile $Q_3$ is the median of the upper half (the last 10 values). The upper half is 4, 4, 4, 4, 5, 5, 5, 5, 6. So $Q_3 = 4.5$
The quartile deviation (QD) is $QD=\frac{Q_3 - Q_1}{2}=\frac{4.5 - 2}{2}=1.25$

Answer:

1. Range: 5
2. Mean absolute deviation: 1.2075
3. Variance: approximately 2.2441
4. Standard deviation: approximately 1.498
5. Quartile Deviation: 1.25