MATHS LESSONStatistical averages

1. Introduction 2. The ’typical’: mode 3. The 'medium': median 4. The 'average': mean 5. Weighted average 6. Summarizing task 7. Exercises 8. Answers and solutions

1.Preliminary problem

It is quite difficult to derive conclusions or hypotheses directly from - occasionally - huge tables or a set of graphs.The charts show the heights of boys and girls in a class of a secondary school.How would it be possible to compare the two sets of data easily?What could - in our case - 'typical', 'medium', and 'average' mean?

2. The 'typical': mode

The animation shows the diagram of three distributions.

(The first shows the number of children in the families of 50 randomly selected students. The second graph displays the distributions of heights of boys in a class. The third shows the distribution of surname-length of 540 pupils of a school in Hungary.)Choose numbers from the first line and put them beside ld put the numbers beside a graph if you think that the number is typical of the data set. Be careful, there are spare numbers. Put these in the waste-bin. 'Typical' is best described as the value that occurs the most often. This is termed mode by mathematicians.When is it necessary or worth using the mode for characterizing a data set?Mode: the data that occur most often in the sample. (If there are more modes, we talk about a set of modes.)Median: the middle element of the increasingly ordered data set.Mean: the sum of the value of the elements divided by the number of elements. If the value ai occurs ni times in a sample, then the average can be formulated as follows:

In this case we talk about the weighted average of the values ai.

3. The 'medium': median

Put the columns in the frame on the broken line in increasing order, then put the red cross on the column representing the medium element.

The median of data can be obtained by arranging the data into increasing order of size and choosing the value that is in the center of the series, that is the one which has as many elements to its right as to its left.Under what conditions can the median of a data set be calculated?

4. The 'average': mean

Here the numeric value of the elements is also important.

The mean of a sample is the sum of the elements divided by the number of elements. Formally:

Calculate the mean of the data on the screen. Check your result.You can write your own data in the frame, the elements separated by commas. Clicking on the OK button, you will see the mean value (m) of the data.

5. Weighted average

Calculating the mean is made easy if the frequency distribution is known.Students of a class received the marks shown in the animation for the latest math-test. Calculate the mean of the class!

You may check your result and watch the procedure of reaching the result by running the animation.This method is called weighted average.You can replace the blue numbers with your own data.

6.Summarizing taskWe have randomly chosen 50 students and asked them how many children there were in their families. We received the following data set:

4 5 2 2 3 4 4 3 5 4 7 3 3 4 2 2 2 2 2 6 3 2 3 3 1 2 3 2 2 6 5 5 3 2 4 4 2 4 1 2 2 2 1 3 3

2 2 4 5 3

Determine the means of the sample!

1.For calculating the mode, create the frequency table of the distribution, grouping the families by the number of children.

2. Put the data in increasing order to obtain the median.

3.Computing the mean is easiest if calculated from the frequency table with the method of weighted average.

7. Exercises

 You are already able to solve the following problems. You can check the correct answers on the Answers page.

1.We conduct a survey in a class, asking students about their favorite drinks. Which average is most suited to characterize the data?

2. You are applying for a job at a company. You can ask the human resource manager about the average of salaries at the company. Which average would you like to know?

3. We collect data about birth dates in a class. The only information important now is the month. Which average represents the resulting data best?

4. We surveyed a class about the amount of time it takes them to get to school. Which average characterizes the data?

5. Students of a class achieved the following results in a test: 5, 10, 25, 25, 25, 30, 30, 30, 30, 35Calculate the averages! Which characterizes the performance best?

6. In a class the average height is 174.5 cm, the average is 168.3 for girls and 179.3 for boys. Are there more boys than girls in the class?

7. Two dices are tossed a thousand times. The sum is written down every time. (Do the experiment using a computer.) Calculate the mean of the data set!

8. Answers

2/1. The main advantage of the mode is that it is easy to compute, and it can be calculated for non-numeric data as well.

3/1. One has to make sure that the data can be ordered.

7/1. Only the mode is suitable for solving this poblem, as the data are non-numeric, and cannot be arranged in order of size, thus have no mean or median.

7/2. The mean could be misleading because of rare but extreme salaries, that of the director for example. Both the mode and the median give more useful information.

7/3. Although the data are numeric, and consequently the mean could be calculated, it does not make much sense (Think about calling the months by their names). Calculating the median is senseless as well. The mode is most suitable for this purpose.

7/4. All averages can be calculated; however, the mean represents the sample best.

7/5. Mean: 24.5, median: 27.5, mode: 30. The mean is rather distorted by the two very poor performances, thus only two of the students performed below average. The mode is also unsuitable to characterize the performance, as only one student performed better than the mode. The median seems to be the best choice, as half the students performed under, and half above the median.

7/6. By the definition of mean: 174.5=(168.3*g+179.3*b)/(g+b). Thus f/l=6.2/4.8, that is: there are more boys than girls in the class.