Measures of Variation - 3.2 - Part 1

Measures of Variation - 3.2

In this section we will explore how to quantify the spread of data.

What are measures of variation?

• Measures of central tendency attempt to take data and summarize them in one number.

• Measures of variation look at how widely dispersed the data is, or, in other words, the spread of the data.

Range

• Tells overall width of data but fails to tell us how much data varies.

Example 1:

A large bakery regularly orders cartons of Maine blueberries. The average weight of the carton is supposed to be 22 ounces. Random samples of cartons from two supplies were weighed. The weight in ounces of the cartons were:

Supplier 1: 17  22  22  22  27

Supplier 2: 17  19  20  27  27 

Results of Example 1:

• Both samples have the same range and average (mean).

• Look back at your answers. Which suppliers blueberries would you probably prefer?

Results of Example 1 cont.

• At a glance we would probably prefer the consistency of Supplier 1’s cartons.

• The question becomes, how do we measure consistency?

Standard deviation:

• a measure that is used to quantify the amount of variation or dispersion of a set of data values (AKA consistency)

•  $s=\sqrt{\frac{\Sigma\left(x-\overline{x}\right)^2}{n-1}}$  

• The formula above is for sample standard deviation

Where did the formula come from?

•  $x-\overline{x}$  tells us how far from the mean a given data value is

• We square the above number to prevent negatives:  $\left(x-\overline{x}\right)^2$  

• We use the sum symbol to total all of the above distances: $\Sigma\left(x-\overline{x}\right)^2$  

• We divide by n-1 (n being the number of data values) to find the average, squared distance a given value is from the mean (also known as variance): $s^2=\frac{\Sigma\left(x-\overline{x}\right)^2}{n-1}$  

• We take the square root to get the standard deviation, which is the average distance a value is from the mean (consistency):  $s=\sqrt{\frac{\Sigma\left(x-\overline{x}\right)^2}{n-1}}$  

Example 2:

Blossom Greenhouse was commissioned to develop an extra large rose for the Rose Bowl Parade. A random sample of blossoms from Hybrid A bushes yielded these diameters (in inches) for mature peak blossoms:

2  3  4  5  6  8  10  10

Example 2 cont.

• Find the mean and standard deviation of the random sample of blossoms from the Hybrid A bush

• To find the mean we add up all the blossom diameters and divide by how many blossoms are in the sample

• 2+3+4+5+6+8+10+10 = 48

• 48/8 = 6

•  $\overline{x}=6$  

Example 2 cont.

Now we will find the standard deviation using our formula: $s=\sqrt{\frac{\Sigma\left(x-\overline{x}\right)^2}{n-1}}$  .

Work your way inside out:
 $\left(2-6\right)^2=16$  
 $\left(3-6\right)^2=9$  
 $\left(4-6\right)^2=4$  
 $\left(5-6\right)^2=1$  
 $\left(6-6\right)^2=0$  
 $\left(8-6\right)^2=4$  
 $\left(10-6\right)^2=16$  
 $\left(10-6\right)^2=16$  

Example 2 cont.

• Now sum all of your $\left(x-\overline{x}\right)^2$  values together

• 16+9+4+1+0+4+16+16=66

• This means $\Sigma\left(x-\overline{x}\right)^2=66$  

• Divide this by n-1

•  $\frac{\Sigma\left(x-\overline{x}\right)^2}{n-1}=\frac{66}{7}=9.428571429$  

Example 2 cont.

• The last step is to take the square root of our variance (9.428....)

•  $\sqrt{9.428571429}=3.07=s$  

• This is the quantified consistency (standard deviation) of the diameter in inches of the roses in the sample from Hybrid Plant A

Standard deviation in the calculator

• stat --> enter --> edit --> put data in L1--> stat--> calc--> 1-var stats--> enter--> 2nd --> L1 --> enter

• The data that pops up are our 1st variable statistics

•  $\overline{x}=$  sample mean

•  $s=$  sample standard deviation

Think about your results from the two different Hybrids of the rose bush? What do they mean?

Multiple Choice

Are Hybrid A or Hybrid B's roses more consistent when it comes to diameter size?

• Hybrid A

• Hybrid B

Conclusion from Example 2

If we are looking for the largest possible rose, we should probably use Hybrid A plants. But be warned, Hybrid A also produces some pretty small roses. If we want to play it safe and ensure most of our blossoms are around 6 inches, we should go with Hybrid B. Since Hybrid B's standard deviation is smaller, its consistency is higher. This means more of its roses will be closer to the mean diameter of 6 inches than Hybrid A's.