A mean is an arithmetic average: add the available numerical values, then divide by how many values you added. It gives you one number to use as a reference point for the group. section 8.8
Finding the center of diamond data
Suppose carat values are recorded in thousandths of a carat. For ten diamonds, the mean carat value is 1,229 (or 1.229 carats), and the mean price is 3,431. A diamond recorded as 300 with a price of 339 sits 929 below the carat mean and 3,092 below the price mean. Subtracting each mean makes it easier to see how carat and price move together. section 8.11
Table 8.9 The diamond dataset and calculations, part 1
|
Carat (x) |
Price (y) |
x – mean of x |
y – mean of y |
|||
| 1 |
300 |
339 |
300 – 1,229 |
-929 |
339 – 3,431 |
-3,092 |
| 2 |
410 |
561 |
410 – 1,229 |
-819 |
561 – 3,431 |
-2,870 |
| 3 |
750 |
2,760 |
750 – 1,229 |
-479 |
2,760 – 3,431 |
-671 |
| 4 |
910 |
2,763 |
910 – 1,229 |
-319 |
2,763 – 3,431 |
-668 |
| 5 |
1,200 |
2,809 |
1,200 – 1,229 |
-29 |
2,809 – 3,431 |
-622 |
| 6 |
1,310 |
3,697 |
1,310 – 1,229 |
81 |
3,697 – 3,431 |
266 |
| 7 |
1,500 |
4,022 |
1,500 – 1,229 |
271 |
4,022 – 3,431 |
591 |
| 8 |
1,740 |
4,677 |
1,740 – 1,229 |
511 |
4,677 – 3,431 |
1,246 |
| 9 |
1,960 |
6,147 |
1,960 – 1,229 |
731 |
6,147 – 3,431 |
2,716 |
| 10 |
2,210 |
6,535 |
2,210 – 1,229 |
981 |
6,535 – 3,431 |
3,104 |
| 1,229 |
3,431 |
|||||
| Means |
Follow the carat and price columns across to see each value compared with its feature's mean. section 8.11
In a simple linear-regression calculation, find the mean of the input values and the mean of the prices separately. Subtract the appropriate mean from each value, then use those differences to calculate the line's slope and intercept. In Python, each mean starts with the same pattern: `sum(values) / len(values)`. section 8.11
A mean is useful, but check what the numbers represent before using it:
- For a missing numerical value, the feature's mean is an easy replacement. It may be misleading if that feature is related to another one: replacing a missing carat value with the mean ignores its relationship with price. It cannot fill a categorical value such as cut or clarity. section 8.8
- For neural-network activations, the mean describes the average result of nodes in a layer. A weight-initialization rule of thumb aims for a mean of zero, alongside consistent variance. section 9.14
