Making comparisons using averages and ranges
Having found a value to represent you data (an average) you can now compare two or more sets of data. However, just comparing the averages can sometimes be misleading.
It can be helpful to know how consistent the data is and you for this by thinking about how spread out the values are. A simple measure of spread is the range.
Range=largest value-smallest value
The larger the range, the more spread out the data is and the less consistent the values are with one another.
Example 1: Two groups of athletes wants to compare their 100 m sprint times.
Each person runs once and records his or her time as shown (in seconds).
a) Calculate the mean 100 m time for each team.
Team Pythagoras:
Mean=$\displaystyle \frac{{14.3+16.6+14.3+17.9+14.1+15.7}}{6}$$\displaystyle =\frac{{92.9}}{6}=15.48\sec $
Team Socrates:
Mean=$\displaystyle \frac{{13.2+16.8+14.7+14.7+13.6+16.2}}{6}$$\displaystyle =\frac{{89.2}}{6}$$\displaystyle =14.87\sec $
b) Which is the smaller mean?
Team Socrates have the smaller mean 100 m time.
c) What does this tell you about the 100 m times for Team Pythagoras in comparison with those for Team Socrates?
The smaller time means that Team Socrates are slightly faster as a team than Team Pythagoras.
d) Calculate the range for each team.
Team Pythagoras range=17.9-14.1=3.8 seconds
e) What does this tell you about the performance of each team?
Team Socrates are slightly faster as a whole and they are slightly more consistent. This suggests that their team performance is not improved significantly by one or two fast individuals but rather all team members run at more or less similar speeds. Team Pythagoras is less consistent and so their mean is improved by individuals.
Calculating averages and ranges for frequency data
So far, the lists of data that you have calculated averages for have been quite small. Once you start to get more than 200 pieces of data is better to collect the data with the same value together and record it in a table. Such a table is known as a frequency distribution table or just a frequency distribution.
Data shown in a frequency distribution table
If you throw a single die 100 times, each of the six numbers will appear several times. You can record the number of times that each appears like this:
Mean
You need to find the total of 100 throws. Sixteen is appeared giving s sub-total of 1 x 16 = 16, thirteen 2s appeared giving a sub- total of 13 x 2 = 26 and so on. You can extend your table to show this:
Tip! You can add columns to a table given to you! It will help you to organize your calculations clearly.
The total of all 100 die throws is the sum of all values in this third column:
=16+26+42+68+95+126=373
So the mean score per throw= $ \displaystyle \frac{{total~~score}}{{total~~number~~of~~throws}}=\frac{{373}}{{100}}=3.73$
Median
There are 100 throws, which is an even number, so the median will be the mean of a middle pair. The first of this middle pair will be found in position $\displaystyle \frac{{100}}{{20}}=50$
The table has placed all the values in order. The first 16 are ones, the next 13 are twos and so on. Notice that adding the first three frequencies gives 16+13+14=43. This means that the first 43 values are 1, 2 or 3. The next 17 values are all 4s, so the 50th and 51st values will both be 4. The mean of both 4s is 4, so this is the median.
Mode
For the mode you simply need to find the die value that has the highest frequency, The number 6 occurs most often (21 times), so 6 is the mode.
Range
The highest and lowest values are known, so the range is 6-1=5
Data organized into a stem and leaf diagram
You can determine averages and the range from stem and leaf diagrams.
Mean
As a stem and leaf diagram shows all the data values, the mean is found by adding all the values and dividing them by the number of values in the same way you would find the mean of any data set.
Median
You can use an ordered stem and leaf diagram to determine the median. An ordered stem and leaf diagram has the leaves for each stem arranged in order from smallest to greatest.
Mode
An ordered stem and leaf diagram allows you see which values are repeated in each row. You can compare these to determine the mode.
Range
In an ordered stem and leaf diagram, the first value and the last value can be used to find the range.
Example 2: The ordered stem and leaf diagram shows the number of customers served at a supermarket checkout every half hour during an 8-hour shift.
a) What is the range of customers served?
The lowest number is 2 and the highest number is 21. The range is 21-2=19 customers.
b) What is the modal number of customers served?
6 is the value that appears most often.
c) Determine the median number of customers served?
There are 16 pieces of data, so the median is the mean of the 8th and 9th values.
$\displaystyle \frac{{(11+13)}}{2}=\frac{{24}}{2}=12$
d) How many customers were served altogether during this shift?
To calculate this, find the sum of all values. Find the total for each row and then combine these to find the overall total.
Row 1: 2+5+5+6+6+6+6=36
Row 2: 11+13+13+15+15+16+17+17=117
Row 3: 21
36+117+21=174 customers in total.
e) Calculate the mean number of customers served every half hour.
$\displaystyle Mean=\frac{{sum~of~data~values}}{{mean~of~data~values}}=\frac{{174}}{{16}}=10.875$ customers for half hour.