Question

3.2 measures of spread: standard deviation and variance
the table below gives the number of hours spent watching tv last week by a sample of 16 children. the values have been sorted. find the minimum, maximum, range, mean, and standard deviation of the following data.
the minimum, maximum, range should be entered as exact values.
round the mean and standard deviation to two decimals places.

|x|
|----|
|13|
|20|
|24|
|27|
|30|
|31|
|34|
|35|
|37|
|38|
|39|
|40|
|43|
|45|
|46|
|50|

min =
max =
range =
mean =
standard deviation =

Explanation:

Step1: Find minimum

The smallest value in the data - set is 13.

Step2: Find maximum

The largest value in the data - set is 50.

Step3: Calculate range

Range = Max - Min. So, Range = 50 - 13 = 37.

Step4: Calculate mean

The formula for the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 16$ and $x_{i}$ are the data - points.
$\sum_{i=1}^{16}x_{i}=13 + 20+24 + 27+30+31+34+35+37+38+39+40+43+45+46+50=532$.
$\bar{x}=\frac{532}{16}=33.25$.

Step5: Calculate standard deviation

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$.
First, calculate $(x_{i}-\bar{x})^{2}$ for each $x_{i}$:
$(13 - 33.25)^{2}=(-20.25)^{2}=410.0625$
$(20 - 33.25)^{2}=(-13.25)^{2}=175.5625$
$(24 - 33.25)^{2}=(-9.25)^{2}=85.5625$
$(27 - 33.25)^{2}=(-6.25)^{2}=39.0625$
$(30 - 33.25)^{2}=(-3.25)^{2}=10.5625$
$(31 - 33.25)^{2}=(-2.25)^{2}=5.0625$
$(34 - 33.25)^{2}=(0.75)^{2}=0.5625$
$(35 - 33.25)^{2}=(1.75)^{2}=3.0625$
$(37 - 33.25)^{2}=(3.75)^{2}=14.0625$
$(38 - 33.25)^{2}=(4.75)^{2}=22.5625$
$(39 - 33.25)^{2}=(5.75)^{2}=33.0625$
$(40 - 33.25)^{2}=(6.75)^{2}=45.5625$
$(43 - 33.25)^{2}=(9.75)^{2}=95.0625$
$(45 - 33.25)^{2}=(11.75)^{2}=138.0625$
$(46 - 33.25)^{2}=(12.75)^{2}=162.5625$
$(50 - 33.25)^{2}=(16.75)^{2}=280.5625$

$\sum_{i = 1}^{16}(x_{i}-\bar{x})^{2}=410.0625+175.5625+85.5625+39.0625+10.5625+5.0625+0.5625+3.0625+14.0625+22.5625+33.0625+45.5625+95.0625+138.0625+162.5625+280.5625 =1523$.

$s=\sqrt{\frac{1523}{16 - 1}}=\sqrt{\frac{1523}{15}}\approx\sqrt{101.5333}\approx10.08$.

Answer:

Min = 13
Max = 50
Range = 37
Mean = 33.25
Standard Deviation = 10.08