Average
Kapitoly: Average, Median, Modus, How to calculate grade point average, Geometric mean, Harmonic mean
The mean is a value often used in statistics that is calculated as the arithmetic average of values.
What is the mean
The average is the classical arithmetic mean of the observed values. We can take this table of wages in CZK as an example:
| Name | wage |
|---|---|
| Tomas | 10 000 |
| Martin | 15 000 |
| Jiří | 18 000 |
| Miroslav | 21 000 |
| Jana | 19 000 |
| Lenka | 25 000 |
| Ondřej | 15 000 |
| Lukas | 45 000 |
| Petra | 29 000 |
| Jan | 85 000 |
| Joseph | 15 000 |
These are the wages of 11 imaginary employees of our imaginary company Matweb s.r.o. What would be the average wage in this company? We would take all the amounts, add them up:
$$ \begin{array}{l} 10 000+ 15 000+ 19 000+ 21 000+ 18 000+ \\ 25 000+ 15 000+ 45 000+ 29 000+ 85 000+ 15 000 = 297 000 \end{array} $$
The total wages paid is 297 000 CZK per month. We get the average wage by dividing this amount by the number of employees, i.e. eleven: 297 000 / 11 = 27 000 The average wage in our company is CZK 27 000. That's not bad, right?
So if we have a set of n values, let's label them x1, x2, …, xn, we get the average value by adding up all the xi and dividing the result by n. We usually label the average value $\overline{x}$. Formula:
$$ \overline{x} = \frac{x_1+x_2+…+x_n}{n} $$
Alternatively, the same formula using sum:
$$ \overline{x} = \frac1n\cdot\sum_{i=1}^nx_i $$
How to calculate the average in Excel
In Czech Excel, the function 'average' is used to calculate the average, in English the function 'average' is used. In the English Excel, we use the average function.
Problems with the average
Some problems you may encounter with the average value:
- The average value does not give the most frequent or most likely value. It is simply an average value. If we were to calculate the average of your grades in school, we might arrive at a number of 2.3, which is not even a valid grade to get in school. But it's an average grade. Similarly, the average wage in our company is 27,000, and nobody actually has that wage.
- The average can give us a bad picture of the data if there are any big extremes in the data. An average wage of CZK 27 000 sounds quite nice, but most employees are far from reaching this wage. Such a high wage is mainly due to one employee, Jan, who takes 85,000, pushing the average much higher. If we fired Jan, the average wage would only be 21,200, which at first glance seems like a less misleading number.