2G Statistics Applications with Decimals

Statistics is the science of collecting and analyzing data to learn about the world around us. Most scientific studies include statistical evidence. It is vital in today’s world to have some training in statistics. The word “statistic” refers to a number calculated from some sample data. In this section, we will be looking at several “statistics”. We will not be learning the meaning behind these calculations yet, but we will instead focus on how to calculate them correctly.

The first statistic we will learn to calculate is the mean average. The mean average is found by adding up all the numbers in the data set and then dividing by how many numbers you have. Remember, the mean is a type of average for the data. However it is not the only type of average. The formula for the mean looks like this: $\text{mean}=\frac{\sum x}{n}$. In this formula the “x” refers to individual numbers in the data set, “n” refers to how many numbers are in the data set, and $\sum$ means to add the data values.

Let’s look at an example of calculating the mean average.

Calculate the mean average weight of cars for the following data. The weights of the cars were measured in tons.

Car Data: 1.60, 1.52, 1.79, 1.44

$$\text{mean}=\frac{\sum x}{n}=\frac{(1.60+1.52+1.79+1.44)}{4}=\frac{6.35}{4}=1.5875$$

We often round statistics like the mean to one more decimal place to the right than is present in the original data. Notice all the numbers in the car data ended in the hundredths place. We will round our answer to one more decimal place to the right than the hundredths place. This means we should round the answer to the thousandths place.

Hence the mean average weight of the cars is approximately 1.588 tons.

The mean is not the only type of average. Another average that is very important in statistics is the median average. To find the median average, you must first put the data in order from smallest to largest. Now count the numbers to the middle. If there are an odd number of values in your data set, there will be a single number in the middle. That middle number is the median. If there are an even number of values in your data set, there will be two numbers in the middle. Add these two middle numbers and divide by 2.

Let’s look at the car weight data again. Remember all the numbers measure weight in tons. This time let’s find the median average.

Car Data: 1.60, 1.52, 1.79, 1.44

First put the data in order from smallest to largest.

1.44, 1.52, 1.60, 1.79

Notice if we count to the middle, there are two numbers in the middle (1.52 and 1.60). Add these two numbers and divide by 2 to get the median.

$$\text{median}=\frac{(1.52+1.60)}{2}=\frac{3.12}{2}=1.56$$

Hence the median average weight of the cars is 1.56 tons.

The next statistic we will learn to calculate is the overall range of the data. The range is not an average. It is a measure of how spread out the data is. To find the range, you must first identify the highest number in the data set (maximum value) and the lowest number in the data set (minimum value).

Range = Max – Min

For an example, let us look at the car data again. Identify the maximum and minimum values in the data and calculate the range.

Car Data: 1.60, 1.52, 1.79, 1.44

Max = 1.79 tons and Min = 1.44 tons

Range = Max – Min = 1.79 – 1.44 = 0.35

So the range of the data is 0.35 tons.

Now do the following examples with your instructor.

Practice Problems Section 2G

For each of the following data sets, calculate the mean average, median average, and the overall range. Remember to round the answers to one more decimal place to the right than the original data has. For amounts given in dollars, round to the nearest cent (hundredths place). For problem 19, the data are given in cents; use the original-data rule and report the mean to three decimal places in cents. Exact median and range values may omit trailing zeros.

1. Exam Scores: 92.6, 73.1, 69.7, 87.4, 94.5

   Check answer

   Mean = 83.46
   Median = 87.4
   Range = 24.8

2. Weights of herbal medicine in grams: 5.7, 5.1, 4.8, 6.2, 4.6, 5.5, 6.3, 5.2

3. Price of movie ticket in dollars (from last ten years):
   $8.58, $8.43, $8.17, $8.13, $7.96, $7.93, $7.89, $7.50, $7.18, $6.88

   Check answer

   Mean = $7.87
   Median = $7.95
   Range = $1.70

4. Height of tree in feet over an eleven year period:
   6.0, 9.5, 13.0, 15.0, 16.5, 17.5, 18.5, 19.0, 19.5, 19.7, 19.8

5. Number of milligrams of medicine given to patients at a hospital:
   5.0, 10.0, 8.5, 7.5, 2.5, 5.5, 6.5, 9.5, 10.5, 12.5, 14.0, 7.5, 3.5, 11.5

   Check answer

   Mean = 8.18
   Median = 8.0
   Range = 11.5

6. Amount of milk in cups from various chocolate cake recipes:
   1.50, 0.75, 1.25, 1.00, 1.75, 2.25, 0.50

7. Area in square kilometers from various national parks in Alaska:
   19185.8, 30448.1, 13044.6, 14869.6, 2711.3, 7084.9, 10601.6, 33682.6

   Check answer

   Mean = 16,453.56 sq km
   Median = 13,957.1 sq km
   Range = 30,971.3 sq km

8. Amount of solar energy in kilowatt hours from solar program at a community college in Northern CA (six months of data, April through September):
   1437.1, 1483.6, 1324.6, 1372.1, 1348.1, 1385.8

9. Amount of solar energy in kilowatt hours from solar program at a community college in Northern CA (six months of data, October through March):
   709.1, 754.5, 537.2, 520.1, 499.4, 1236.5

   Check answer

   Mean = 709.47 kilowatt hours
   Median = 623.15 kilowatt hours
   Range = 737.1 kilowatt hours

10. The amount of money in a savings account in dollars:
    $500.00, $638.64, $751.83, $960.30

11. The exam scores on a philosophy test: 84.3, 91.2, 94.6, 81.3

    Check answer

    Mean = 87.85
    Median = 87.75
    Range = 13.3

12. The number of tons of trash brought by trucks to a trash dump:
    5.7, 3.2, 7.6, 9.1, 8.2, 6.9

13. The cost of various laptop computers: $866.23, $1524.90, $582.35, $ 799.63

    Check answer

    Mean = $943.28
    Median = $832.93
    Range = $942.55

14. Height of women on a basketball team in inches: 72.1, 69.5, 75.2, 77.3, 65.8

15. Weight of men on a professional football team in pounds:
    255.5, 310.4, 269.0, 247.5, 330.2, 239.7

    Check answer

    Mean = 275.38 pounds
    Median = 262.25 pounds
    Range = 90.5 pounds

16. Time (in minutes) to run a mile for athletes on a cross country track team:
    5.25, 4.85, 4.30, 5.05, 4.52

17. Price of various bass guitars in dollars:
    $399.99, $799.50, $849.99, $2325.00, $529.60, $1850.49

    Check answer

    Mean = $1125.76
    Median = $824.75
    Range = $1925.01

18. Length of various two-by-fours at a lumber yard in meters.

    1.75, 1.50, 1.35, 2.25, 2.75, 0.80, 1.25

19. Prices in cents of a pound of Colombian coffee during various months:
    82.07, 91.55, 103.24, 101.79, 99.14, 96.01, 92.45

    Check answer

    Mean = 95.179 cents
    Median = 96.01 cents
    Range = 21.17 cents

20. The average temperature in Fargo, North Dakota during winter in degrees Fahrenheit:

    9.1, 12.6, 0.4, 3.1, 7.4, 18.2, 5.4, 1.7, 6.9

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.