The most common summary statistics are those that use one or more numbers to summarise a data collection. In some literature, the mean, median, and mode are all referred to as “averages,” however it is preferable to use their scientific names to avoid misunderstanding. Because there is a geometric mean, a harmonic mean, a geometric median, and so on, it is not always clear from context which one you are referring to. Adding “arithmetic” also helps. Here are some quick definitions, as well as when each is most suited, as well as examples depending on the numbers: 1, 2, 3, 3, 5, 10, 18, 20, 20, 28, 32, 33, 34, 35, 36, 37, 38, 39
The arithmetic mean is calculated by multiplying the total of all values by their count. According to our Mean, Median, and Mode calculator, the mean of the above collection is 10. The mean is an unbiased statistic since the sum of the differences between the mean and each element of a set equals zero. It also has the lowest root mean squared error (RMSE) and is hence the set’s single best predictor.
When you wish to offer a value that, if you used it to predict about any value in the set, you would be closest to each element on average, you should use the mean.The mean is a simple concept that is sometimes misinterpreted, especially when applied to skewed or large-range distributions.
The median is the point at which an ordered set of data is divided in half. The above set’s median, which is ranked from lowest to highest, is 5. Half of the items will have a value lower than the median, while the other half will have a value higher. The arithmetic mean of the two elements around the mid-point is commonly used to calculate the median for a set with an even number of elements.
When we wish to talk about halves, we utilise the median, which is a reference point from which 50% of the values fall above or below. It is thus employed in income and revenue statistics, as well as occasionally in height and weight statistics – for example, in CDC data on adults aged 0 to 21.
In a set, the mode is the most often occurring value. The example set’s mode is 3, which appears twice, whereas every other value appears just once. It’s possible for a set to have more than one mode; it might be bimodal or even multimodal. For example, if you add “20” to the aforementioned set, it will have two modes: 3 and 20.
The mode is utilised in elections and other types of voting, where counting is essentially determining the mode of the votes cast. If you want to guess the exact value of a randomly picked element from the set, a mode is likewise your best bet. You can surely use the aforementioned mode calculator to rank the numbers by frequency and display the set’s mode (or modes).
Simply said, the range is the difference between the lowest and highest value in a set. They are instantly obvious if the set is sorted; otherwise, you must order the set in ascending order and then take the first and last values. The range in the given sample set is 1-28.
When you want to know the minimum and maximum conceivable values, values beneath which and above which there are no other values, the range is the proper statistic to calculate using our range calculator above. In population statistics, disease symptoms, and other areas, these are frequently extreme cases.
Let’s look at some hypothetical annual wage data for three companies with 17 employees each: A, B, and C. This is what the raw data looks like:
Let’s compare the three statistics for each of the three companies:
We can observe that the arithmetic average ($57,705.88) is the same for all three, but the median (from $23,000 to $49,000) and mode differ dramatically. The median is equal to the mode for two of the companies. We would consider the median to be the most ideal number in many circumstances.
The same data is summarised using the arithmetic mean and the min-max range as follows:
Having the range provides you an idea of where the density of the distribution falls, and having the average gives you an idea of where the density of the distribution falls. (With the help of our mean and range calculator)
The preceding is only one example. There are many examples of data from completely distinct distributions having the same summary statistics, so always request the raw data or at the very least a plot showing the entire distribution of the data, together with the statistics that are regarded to be the most relevant.
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