Dispersion

Every day we meet numbers about the world around us: the marks of a class in a test, the rainfall of a district in a month, the number of tourists who visit Pokhara in a year. Numbers like these, gathered for a purpose, are called data. A long list of numbers on its own says very little. Someone has to collect the numbers carefully, arrange them, work on them and then explain what they show.

Each of the four words in that definition is a separate job. Collection means gathering the data. Analysis means working on it, for example by finding an average. Interpretation means saying what the results mean. Presentation means showing the data clearly in a table, a graph or a diagram, so that other people can understand it quickly.

Statistics is as old as human civilisation. In its early days it was used mainly for running the state. A ruler needed to know how many people lived in the kingdom and what resources it had, so that taxes could be collected and plans could be made. In ancient Babylonia, Egypt and China, statistics was used to count the population and the resources of the land. From there the subject grew step by step.

• John Graunt (1620 to 1674) analysed the death rates of London and was the first person to present data in a scientific way.
• In the twentieth century, Sir Ronald A. Fisher laid the foundation of modern statistics. For this reason he is called the father of statistics.
• Later, Karl Pearson and William Sealy Gosset (1876 to 1937) made further contributions to the subject.
• In today's digital age, computers and big data have made statistics more powerful and far wider in use. It is now used in almost every field, including science and technology.

Big data means collections of data so large that only computers can handle them, such as a record of every mobile phone payment made in a country in a single day. Studying data of that size would be impossible by hand, but a computer can find its average or its pattern in seconds.

When we study any set of data, we usually begin by finding a single value that stands for the whole set. You have met three such values already: the mean, the median and the mode.

• Mean: the sum of all the observations divided by the number of observations.
• Median: the middle value when the observations are arranged in order.
• Mode: the value that occurs most often.

The mean of individual observations is found with this formula:

\[ \bar{x} = \frac{\sum x}{n} \]

Here \( \sum x \) is the sum of all the observations and \( n \) is the number of observations. A measure of central tendency turns a large amount of information into one simple and meaningful number. If the mean mark of a class of 40 students is 62, we get a quick picture of the class without reading all 40 marks.

A measure of central tendency cannot always tell the whole story about a set of data. Two or more sets can have exactly the same central value and still be distributed very differently. The way the numbers are spread out can differ a great deal. Look at the marks obtained by five students in each of three different classes.

We find the mean of each group, one at a time.

Step 1: Adding the five marks of Group A:

\[ 48 + 49 + 50 + 51 + 52 = 250 \]

Step 2: Dividing this total by the number of students, which is 5:

\[ \bar{x}_A = \frac{250}{5} = 50 \]

Step 3: Adding the five marks of Group B:

\[ 40 + 45 + 50 + 55 + 60 = 250 \]

Step 4: Dividing by 5 to get the mean of Group B:

\[ \bar{x}_B = \frac{250}{5} = 50 \]

Step 5: Adding the five marks of Group C:

\[ 20 + 35 + 50 + 65 + 80 = 250 \]

Step 6: Dividing by 5 to get the mean of Group C:

\[ \bar{x}_C = \frac{250}{5} = 50 \]

All three groups have the same mean, 50. Yet one look at the marks shows the groups are not alike at all. In Group A nobody scored below 48 or above 52. In Group C one student scored only 20 while another scored 80.

Marking each mark as a dot on a number line from 0 to 100 makes the difference easy to see. The mean, 50, sits at the same place on all three lines. What changes is how the dots gather around it.

• (a) Group A has low variation. All its marks are very close to the mean of 50, which shows that the students of this group achieved almost the same.
• (b) Group B has moderate variation. Its marks are spread out at a fair distance from the mean.
• (c) Group C has high variation. Its marks are spread far apart, which shows a big gap between the achievements of its students.

We can put numbers on what the dots show by asking how far each mark lies from the mean of 50.

The means are identical, but the distances grow from Group A to Group C. What makes these three sets of data different from one another is their degree of spread, or variation.

The spread we have just seen has a name in statistics.

When the observations huddle close to the centre, as in Group A, the dispersion is low. When they are scattered far from the centre, as in Group C, the dispersion is high. If every observation were the same, for example 50, 50, 50, 50 and 50, no value would differ from any other, and there would be no dispersion at all.

The same idea appears outside the classroom. Two batsmen, P and Q, played five cricket matches each and scored these runs.

Step 1: Adding the runs of Batsman P:

\[ 38 + 40 + 42 + 39 + 41 = 200 \]

Step 2: Adding the runs of Batsman Q:

\[ 0 + 90 + 5 + 100 + 5 = 200 \]

Step 3: Dividing each total by the 5 matches played, first for P:

\[ \bar{x}_P = \frac{200}{5} = 40 \]

Step 4: Dividing Q's total by 5 in the same way:

\[ \bar{x}_Q = \frac{200}{5} = 40 \]

Both batsmen average 40 runs. But Batsman P never strays far from 40: his lowest score is 38 and his highest is 42, only 2 runs away from his mean. Batsman Q scored 0 in one match and 100 in another, and 100 is 60 runs above his mean. P has low dispersion and Q has high dispersion. A captain who needs a steady innings every match would choose P, and the average alone could never have told him that.

A measure of dispersion helps us in the following ways.

• (a) To find out how consistent or how variable a set of data is.
• (b) To compare the reliability of two or more sets of data.
• (c) To check how well a measure of central tendency represents the data.
• (d) To gain deeper insight for making decisions in fields such as education, business, economics and science.

Each of these uses can be seen in the examples above. Batsman P is consistent and Batsman Q is variable. P's scores are more reliable, because we can expect something close to 40 from him next time. The mean of 50 represents Group A very well, since every mark lies within 2 of it, but it represents Group C poorly, since most marks there lie 15 or 30 away from it. And in real decisions, a farmer choosing between two kinds of seed with the same average harvest would prefer the one whose harvest changes least from year to year, while a factory wants every bolt it makes to be nearly the same length.

So whenever you describe a set of data, give two facts: where its centre lies, and how widely the values spread around that centre. Groups A and C share the first fact completely and differ completely in the second.