With large and/or confusing amounts of data, it is useful to have a few values that describe the amount of data as appropriately as possible. In descriptive (descriptive) statistics, data series are summarized, for example, using measures of location and measures of dispersion.
With the term Average is in this article that arithmetic means meant. The arithmetic mean is one measure of location and is used to describe the middle of a data set.
Sometimes the mean is also used as a generic term, similar to the position measure. Then, for example, the arithmetic mean, the median or the mode are summarized as mean values.
Arithmetic mean – definition
Of the Average is the most common and well-known measure of position in everyday life.
Of the Average is calculated by dividing the sum of all data by the number of data:
.
The mean will also arithmetic mean called. Colloquially, the mean is also often referred to as Average designated. The mean can be with a x with a dash (), called «x across».to be discribed.
The mean is used, for example, when calculating the grade point average for a school assignment. So you already know him.
But now let’s look at another example.
Task 1
After the summer holidays, a class determines how many days were spent abroad during the holidays. The results are noted in the following table:
Calculate the mean.
solution
A total of 15 pieces of data were collected:
On average, each student has been abroad for 11 days.
Calculate mean
The «raw data» (unprocessed data) is not always available. For this reason, mean values sometimes have to be determined from frequency distributions, which are displayed, for example, in bar, column, or pie charts.
One frequency distribution Specifies how often a specific value occurs in the original list.
Calculate the mean using the absolute frequency distribution
Given the absolute frequencies, you can calculate the arithmetic mean.
Calculating the arithmetic mean from an absolute frequency distribution:
To do this, you multiply the absolute frequencies by the corresponding number of this value and add these products. You then divide that by the sum of the absolute frequencies.
The absolute frequencies indicate how often the corresponding characteristic occurs in the original list.
Formulas often sound complicated when you see them written down in such general terms. But if you look at the example, it becomes clear how it works.
exercise 2
The number of siblings of seventh graders was recorded. You can see the results in the bar chart.
Figure 1: Bar chart – number of siblings of seventh grade students
Then calculate the arithmetic mean.
If you don’t know exactly what a column chart is, you can read it again in the dedicated article.
solution
- To calculate the numerator, you always have to take the product of the value and the number and add these products together.
- To calculate the denominator, you need to calculate the count of all values.
On average, a student in this seventh grade has 1.7 siblings.
Calculate the mean using the relative frequency distribution
You can not only determine the mean from an absolute frequency distribution, you can also calculate it if relative frequencies are given.
the relative frequency distribution Indicates the proportion of a specific feature in the totality of the data.
Relative frequencies are often given as percentages.
Calculating the arithmetic mean from a relative frequency distribution:
The arithmetic mean can be calculated by multiplying the values by their relative frequencies and then adding these products.
The next example will show you how to calculate the arithmetic mean given percentages or relative frequencies.
task 3
The pie chart shows the distribution of grades for the last homework.
Figure 2: Pie chart – grade distribution of a school assignment
Calculate the grade point average for the class.
If you’re not sure what a pie chart is, you can read about it in the dedicated article.
solution
Given relative frequencies, calculate the mean by adding the products of value and relative frequency.
Danger! You need to convert the percentages to decimals first. This works by shifting the decimal point 2 places to the left. For example:
The grade point average in the school assignment is 3.2.
Special averages
The arithmetic mean is not always the best way to describe data. That’s why there are other mean values.
Weighted mean
With the arithmetic mean, all data are weighted equally. However, if the values in a data series are not to be weighted equally, but some may have a higher or lower weight than others, the weighted mean and not the arithmetic mean is used.
You already know this type of calculation, because your grade in a school subject is made up of schoolwork grades, oral grades and Exen. The school assignment is weighted higher (double) than the other grades.
Of the weighted meanrt or the weighted arithmetic mean is calculated by multiplying the data values by their weight and dividing the sum of these products by the sum of the weights:
.
Here are the data and the associated weights.
Consider this example where the data of the data set are weighted differently.
task 4
The average oil consumption in Finland is about 5,925 kg per capita, in Norway 5,820 kg per capita and in Sweden, 5,102 kg per capita.
If you had to calculate the average oil consumption of the three countries, you could take the arithmetic mean of the three values. However, this does not reflect the real situation well because the countries have different numbers of inhabitants.
It is therefore better to weight the individual values with the number of inhabitants.
- Norway: 5 million inhabitants
- Sweden: 10 million inhabitants
- Finland: 5 million inhabitants
Calculate the mean oil consumption of the three countries.
solution
You now have to calculate the average weighted with the population figures.
On average, an inhabitant of the three countries consumes 5487 kg,
Moving Average
The moving average is used when measuring data as a function of time. With this method, the arithmetic mean is always formed over the same number of values. The value at a specific measurement time and a specific number of values before and after are always summarized. The arithmetic mean is formed from the values within this sliding measurement window.
The aim is to even out fluctuations and thus smooth the curve of the measured data.
Consider the blue curve that shows the monthly precipitation for Germany in 2021.
Figure 3: Precipitation in Germany in 2021 given monthly and with a moving averageSource: de.statista.com
The moving average was used for the orange, smoothed curve. Each orange dot is the mean of three readings.
- The first orange value is therefore the mean of the values for January, February and March
- The second orange point is then the mean of the values for February, March and April
- …
Difference between median and mean
In addition to the mean, there are other measures such as the median. Both the median and the arithmetic mean are easy ways to summarize a data set.
Of the median is the value in an ordered record right in the middle lies. Ordered means that the values are written in order of magnitude.
Sometimes the median is also called central value designated. , or is often used as a symbol.
When calculating the arithmetic mean, all the data in the data set are taken into account. Therefore, the mean reacts strongly to outliers, i.e. to values that are significantly smaller or larger than the others. It is also said that the arithmetic mean not robust to outliers is.
When determining the median, on the other hand, only one or two pieces of data from the data series are included. The median is therefore robust to outliers, since its value does not change due to a particularly large or small value.
Let’s take a look at this example where we want to determine the median and the arithmetic mean.
task 5
A pharmaceutical company wants to test whether the new drug is better for lowering blood pressure than the old one. To do this, they measured the blood pressure before and after taking the drug. The table shows the level of blood pressure reduction (in mmHg).
old drug
new drug
15
17
4
7
9
6
19
14
17
15
6
17
51
17
14
18
9
15
A drug is considered successful in lowering blood pressure if blood pressure is reduced by at least 15 mmHg.
Determine the median and mean and decide whether the new drug is better than the old one.
solution
old drug
new drug
Calculate mean
To determine the median, the data set is sorted and then the mean value is read.
As you can see, the mean is higher for the old drug than for the new drug. The median, on the other hand, is the opposite. So which drug is better?
If you remember, a drug was considered successful if blood pressure was reduced by at least 15 mmHg.
- Because the median of the old drug is 14 mmHg, the drug is ineffective in at least half of the cases. In fact, the drug was only effective in 4 out of 9 cases.
- The median of the new drug, however, is 15 mmHg, so that the blood pressure reduction was successful in at least half of the cases. In fact, the drug was effective in 6 out of 9 attempts.
So why is the mean of the old drug so much higher than that of the new drug?
This is because the value 51 mmHg is so far removed from the other values and thus strongly influences the arithmetic mean. But just because the drug worked very well in one case doesn’t make it a better drug overall. In addition, one could consider whether the reading might be inaccurate since it deviates so far from the others.
Overall, it can be said that the new drug is better suited for lowering blood pressure than the old one.
Mean – The most important
- Of the Average or that arithmetic means is the average value in a data set.
- Calculate the mean of the raw data:
- Calculate the mean from a absolute frequency distribution:
- Calculate the mean from a relative frequency distribution:
- The mean has one low robustness to outliers.
- Of the weighted meanrt is calculated by multiplying the data values by their weights and dividing the sum of these products by the sum of the weights:
- Moving Average: The moving average is taken when data depends on the…