Question

name: ____________ score: ____________
section: ____________ date: ____________
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 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, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6

Step2: Calculate the range

Range = Maximum - Minimum
Maximum = 6, Minimum = 1
Range = 6 - 1 = 5

Step3: Calculate the mean

$\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$
$\sum_{i=1}^{20}x_{i}=1\times4 + 2\times3+3\times3 + 4\times4+5\times4+6\times1=70$
$n = 20$
$\bar{x}=\frac{70}{20}=3.5$

Step4: Calculate the absolute - deviation

$|x_{i}-\bar{x}|$ for each $x_{i}$:
For $x = 1$, $|1 - 3.5|=2.5$ (4 times)
For $x = 2$, $|2 - 3.5| = 1.5$ (3 times)
For $x = 3$, $|3 - 3.5|=0.5$ (3 times)
For $x = 4$, $|4 - 3.5| = 0.5$ (4 times)
For $x = 5$, $|5 - 3.5|=1.5$ (4 times)
For $x = 6$, $|6 - 3.5| = 2.5$ (1 time)

Step5: Calculate the mean absolute deviation (MAD)

$MAD=\frac{\sum_{i = 1}^{n}|x_{i}-\bar{x}|}{n}$
$\sum_{i = 1}^{20}|x_{i}-\bar{x}|=2.5\times4+1.5\times3 + 0.5\times3+0.5\times4+1.5\times4+2.5\times1$
$=10 + 4.5+1.5 + 2+6 + 2.5=26.5$
$MAD=\frac{26.5}{20}=1.325$

Step6: Calculate the variance

$s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}$
$(x_{i}-\bar{x})^{2}$ for each $x_{i}$:
For $x = 1$, $(1 - 3.5)^{2}=6.25$ (4 times)
For $x = 2$, $(2 - 3.5)^{2}=2.25$ (3 times)
For $x = 3$, $(3 - 3.5)^{2}=0.25$ (3 times)
For $x = 4$, $(4 - 3.5)^{2}=0.25$ (4 times)
For $x = 5$, $(5 - 3.5)^{2}=2.25$ (4 times)
For $x = 6$, $(6 - 3.5)^{2}=6.25$ (1 time)
$\sum_{i = 1}^{20}(x_{i}-\bar{x})^{2}=6.25\times4+2.25\times3+0.25\times3+0.25\times4+2.25\times4+6.25\times1$
$=25+6.75 + 0.75+1+9+6.25=48.75$
$s^{2}=\frac{48.75}{19}\approx2.566$

Step7: Calculate the standard deviation

$s=\sqrt{s^{2}}=\sqrt{2.566}\approx1.602$

Step8: Calculate quartiles

$n = 20$
The first - quartile $Q_{1}$:
The position of $Q_{1}=\frac{n + 1}{4}=\frac{20+1}{4}=5.25$
$Q_{1}=1+(2 - 1)\times0.25 = 1.25$
The third - quartile $Q_{3}$:
The position of $Q_{3}=\frac{3(n + 1)}{4}=\frac{3\times(20 + 1)}{4}=15.75$
$Q_{3}=4+(5 - 4)\times0.75 = 4.75$
Quartile deviation $QD=\frac{Q_{3}-Q_{1}}{2}=\frac{4.75 - 1.25}{2}=\frac{3.5}{2}=1.75$

Answer:

1. Range: 5
2. Mean absolute deviation: 1.325
3. Variance: approximately 2.566
4. Standard deviation: approximately 1.602
5. Quartile deviation: 1.75