*Deviations
You can draw many lines through the central point, including a horizontal one.

Why would this not be a sensible line?

The best trend line is one that takes account of all the points fairly.

One way to check this is to find the vertical distances of individual points from the line drawn. These distances are called DEVIATIONS.

The deviations should be as small as possible.

*Considering Deviations
Figure 2 shows the cinema attendance figures of Table 3 and a possible trend line. Notice that it passes through the central point. Some of the deviations are marked in. All are given in Table 4.

Table 4 - Deviations of cinema attendances given from the trend line drawn (Millions).

Figure 2 - Cinema attendances (in millions), 1965-1974, with a possible trend line.

The deviation is the difference between the actual attendance and the attendance obtained from the trend line. Notice that if a point is above the line, it has been given a positive (+) deviation and if below a negative deviation.

1. Make a copy of Table 4, leaving the last two columns blank, or use Table 8 on page R1.
2. Using the line you drew in Section Cl, read off values to complete column 3.
3. Work out the deviations, and complete column 4.
4. How can these deviations he used to check which is the best line?

One possible suggestion is to add all the deviations together.

5. What would this give for Table 4?
6. What happens if you do this in your own table?
7. Can you give a reason? If so, write it down.
8. Suggest other ways these deviations could be used. In each case check the effect of the suggestion on the figures in Table 4 and also on your own figures.

*The Least Squares Method
The usual method is this. Square the deviations and find their total.

A line close to the points will have a small sum. A line far from the points will have a large sum. The line with the smallest value is best.

Table 5 shows the squared deviations obtained from Table 4.

Remember the deviations are in millions (10⁶), so the squared deviations are in millions of millions (10¹²).

1. Make a similar table using your deviations, or complete Table 9 on page R1.
2. Compare your total with that in Table 5. Which is the smaller?

Table 5 - Deviations and squared deviations.

3. Compare your total with those of other people in the class.
   What is the smallest total?
   Have you checked their arithmetic?
   Does the line that gives the smallest total seem to be a good fit?

Tables similar to 4 and 5 can be drawn for any set of data.

4. *Repeat the process for the line you drew in Section C2. What is the smallest total this time?

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