Question

name: ____________ score: __________ section: __________ date: ____________ the number of incorrect answers on a true - or - false mathematics proficiency test for a 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, 1, 2, 2, 2, 3, 3, 3, 4, 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

$\bar{x}=\frac{1\times4 + 2\times3+3\times3 + 4\times5+5\times4+6\times1}{20}=\frac{4 + 6+9 + 20+20+6}{20}=\frac{65}{20}=3.25$

Step4: Calculate the absolute - deviation

$|x_1-\bar{x}|,|x_2 - \bar{x}|,\cdots,|x_{20}-\bar{x}|$
For $x = 1$: $|1 - 3.25|=2.25$ (4 times)
For $x = 2$: $|2 - 3.25| = 1.25$ (3 times)
For $x = 3$: $|3 - 3.25|=0.25$ (3 times)
For $x = 4$: $|4 - 3.25| = 0.75$ (5 times)
For $x = 5$: $|5 - 3.25|=1.75$ (4 times)
For $x = 6$: $|6 - 3.25| = 2.75$ (1 time)
The sum of absolute - deviations is $4\times2.25+3\times1.25 + 3\times0.25+5\times0.75+4\times1.75+1\times2.75$
$=9+3.75 + 0.75+3.75+7+2.75=27$
The mean absolute deviation (MAD) is $\frac{27}{20}=1.35$

Step5: Calculate the variance

The variance $s^{2}=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}$
$(x_1-\bar{x})^2,(x_2 - \bar{x})^2,\cdots,(x_{20}-\bar{x})^2$
For $x = 1$: $(1 - 3.25)^2=5.0625$ (4 times)
For $x = 2$: $(2 - 3.25)^2 = 1.5625$ (3 times)
For $x = 3$: $(3 - 3.25)^2=0.0625$ (3 times)
For $x = 4$: $(4 - 3.25)^2 = 0.5625$ (5 times)
For $x = 5$: $(5 - 3.25)^2=3.0625$ (4 times)
For $x = 6$: $(6 - 3.25)^2 = 7.5625$ (1 time)
The sum of squared - deviations is $4\times5.0625+3\times1.5625+3\times0.0625+5\times0.5625+4\times3.0625+1\times7.5625$
$=20.25+4.6875 + 0.1875+2.8125+12.25+7.5625=47.75$
The variance $s^{2}=\frac{47.75}{19}\approx2.5132$

Step6: Calculate the standard deviation

The standard deviation $s=\sqrt{s^{2}}=\sqrt{2.5132}\approx1.5853$

Step7: Calculate the quartile deviation

First, find the first quartile $Q_1$ and the third quartile $Q_3$.
Since $n = 20$, the position of $Q_1$ is $\frac{n + 1}{4}=\frac{20+1}{4}=5.25$
$Q_1$ is the value between the 5th and 6th ordered data values. The 5th value is 2 and the 6th value is 2, so $Q_1 = 2$
The position of $Q_3$ is $\frac{3(n + 1)}{4}=\frac{3\times(20 + 1)}{4}=15.75$
$Q_3$ is the value between the 15th and 16th ordered data values. The 15th value is 4 and the 16th value is 5, so $Q_3=\frac{4 + 5}{2}=4.5$
The quartile deviation $QD=\frac{Q_3 - Q_1}{2}=\frac{4.5 - 2}{2}=1.25$

Answer:

1. Range: 5
2. Mean absolute deviation: 1.35
3. Variance: approximately 2.5132
4. Standard deviation: approximately 1.5853
5. Quartile deviation: 1.25