Question

72, 83, 90, 78, 85, 83, 77, 93, 80, 75
calculate the:
mean =
median =
mode =
range =
standard deviation (round to two decimal places)

Explanation:

Step1: Count the number of data points

We have the data set: 72, 83, 90, 78, 85, 83, 77, 93, 80, 75. The number of data points \( n = 10 \).

Step2: Calculate the Mean

The mean \( \bar{x} \) is calculated by the formula \( \bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n} \).
First, sum all the data points:
\( 72+83 + 90+78+85+83+77+93+80+75\)
\(=72 + 83=155\); \(155+90 = 245\); \(245+78=323\); \(323+85 = 408\); \(408+83=491\); \(491+77 = 568\); \(568+93=661\); \(661+80 = 741\); \(741+75=816\)
Then, \( \bar{x}=\frac{816}{10}=81.6 \)

Step3: Calculate the Median

First, sort the data set in ascending order: 72, 75, 77, 78, 80, 83, 83, 85, 90, 93.
Since \( n = 10 \) (even), the median is the average of the \( \frac{n}{2}\)-th and \( (\frac{n}{2}+ 1)\)-th values.
\( \frac{n}{2}=\frac{10}{2}=5 \)-th value is 80, \( \frac{n}{2}+1 = 6 \)-th value is 83.
Median \(=\frac{80 + 83}{2}=\frac{163}{2}=81.5\)

Step4: Calculate the Mode

The mode is the value that appears most frequently. In the data set, 83 appears twice, and all other values appear once. So the mode is 83.

Step5: Calculate the Range

The range is the difference between the maximum and minimum values.
Maximum value \( = 93 \), Minimum value \( = 72 \)
Range \(=93 - 72=21\)

Step6: Calculate the Standard Deviation

The formula for sample standard deviation (since we have a sample of data, if it were population, we would divide by \( n \) instead of \( n - 1 \); here, since it's a set of values (maybe a sample), we use \( n-1 \)) is \( s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}} \)
First, calculate \( (x_{i}-\bar{x})^{2} \) for each \( x_{i} \):

• For \( x = 72 \): \( (72 - 81.6)^{2}=(-9.6)^{2}=92.16 \)
• For \( x = 75 \): \( (75 - 81.6)^{2}=(-6.6)^{2}=43.56 \)
• For \( x = 77 \): \( (77 - 81.6)^{2}=(-4.6)^{2}=21.16 \)
• For \( x = 78 \): \( (78 - 81.6)^{2}=(-3.6)^{2}=12.96 \)
• For \( x = 80 \): \( (80 - 81.6)^{2}=(-1.6)^{2}=2.56 \)
• For \( x = 83 \): \( (83 - 81.6)^{2}=(1.4)^{2}=1.96 \) (two times, so total \( 2\times1.96 = 3.92 \))
• For \( x = 85 \): \( (85 - 81.6)^{2}=(3.4)^{2}=11.56 \)
• For \( x = 90 \): \( (90 - 81.6)^{2}=(8.4)^{2}=70.56 \)
• For \( x = 93 \): \( (93 - 81.6)^{2}=(11.4)^{2}=129.96 \)
• For \( x = 75 \): already calculated as 43.56 (wait, no, 75 is a separate data point, we already did 75 above)

Now sum all these squared differences:
\(92.16+43.56+21.16+12.96+2.56 + 3.92+11.56+70.56+129.96\)
Let's calculate step by step:
\(92.16+43.56 = 135.72\); \(135.72+21.16=156.88\); \(156.88+12.96 = 169.84\); \(169.84+2.56=172.4\); \(172.4+3.92 = 176.32\); \(176.32+11.56=187.88\); \(187.88+70.56 = 258.44\); \(258.44+129.96=388.4\)

Now, \( n-1=9 \), so \( \frac{\sum(x_{i}-\bar{x})^{2}}{n - 1}=\frac{388.4}{9}\approx43.16\)
Then, standard deviation \( s=\sqrt{43.16}\approx6.57 \)

Answer:

• Mean: \( 81.6 \)
• Median: \( 81.5 \)
• Mode: \( 83 \)
• Range: \( 21 \)
• Standard Deviation: \( 6.57 \) (rounded to two decimal places)