In statistics the mode of a set of numbers is the value that appears most frequently within the sample. A dataset does not necessarily have only one fashion; if two or more values are "destined" to be the most common, then we speak of a bimodal or multimodal set, respectively. In other words, all the most common values are the fashions of the sample. Read on for more details on how to determine the fashion of a set of numbers.
Steps
Method 1 of 2: Finding the Mode of a Data Set
Step 1. Write down all the numbers that make up the set
The mode is usually calculated from a set of statistical points or a list of numerical values. For that reason, you need a data group. Calculating fashion in mind is not easy at all, unless it is a rather small sample; therefore in most cases it is advisable to write by hand (or type on the computer) all the values that make up the set. If you are working with pen and paper, just list all the numbers in sequence; if you are using computer, it is best to set up a spreadsheet to outline the process.
It is easier to understand the process with an example problem. In this section of the article, we consider this set of numbers: {18; 21; 11; 21; 15; 19; 17; 21; 17}. In the next few steps, we will find the sample fashion.
Step 2. Write the numbers in ascending order
The next step is usually to rewrite the data from the smallest to the largest. Even if it is not a strictly essential procedure, it makes the calculation much easier, because the identical numbers will be found grouped. If it is a very large sample, however, this step is essential, because it is practically impossible to remember how many times a value occurs and you could make mistakes.
- If you're working with pencil and paper, rewriting the data will save you time in the future. Analyze the sample looking for the smallest value and, when you find it, cross it off the initial list and rewrite it in the new sorted set. Repeat the process for the second smallest number, third, and so on, making sure to rewrite the number each time it occurs in the set.
- If you are using the computer, you have a lot more possibilities. Several calculation programs allow you to reorder a list of values from the largest to the smallest with a few simple clicks.
- The set considered in our example, once rearranged, will look like this: {11; 15; 17; 17; 18; 19; 21; 21; 21}.
Step 3. Count the number of times each number repeats
At this point you need to know how many times each value appears within the sample. Look for the number that occurs most frequently. For relatively small sets, with the data reordered, it is not difficult to recognize the largest "cluster" of identical values and to count how many times the data repeats.
- If you are using pen and paper, then make a note of your calculations by writing next to each value how many times this repeats. If you are using a computer, you can do the same by noting the frequency of each data in the adjacent cell or by using the program's function that counts the number of repetitions.
- Let's consider our example again: ({11; 15; 17; 17; 18; 19; 21; 21; 21}), 11 occurs once, 15 once, 17 twice, 18 once, the 19th one and the 21 three times. So we can say that 21 is the most common value in this set.
Step 4. Identify the value (or values) that occurs most frequently
When you know how many times each piece of data is reported in the sample, find the one that has the most repetitions. This represents the fashion of your ensemble. Note that there can be more than one fashion. If two values are the most common, then we speak of a bimodal sample, if there are three frequent values, then we speak of a trimodal sample and so on.
- In our example ({11; 15; 17; 17; 18; 19; 21; 21; 21}), since 21 occurs more times than the other values, then you can say that 21 is fashion.
- If another number besides 21 had occurred three times (for example if there had been another 17 in the sample), then 21 and this other number would both have been fashionable.
Step 5. Don't confuse fashion with mean or median
These are three statistical concepts that are often discussed together because they have similar names and because, for each sample, a single value can simultaneously represent more than one. All this can be misleading and lead to error. However, regardless of whether or not the fashion of a group of numbers is also the mean and the median, you must remember that these are three completely independent concepts:
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The mean of a sample represents the mean value. To find it, you have to add all the numbers together and divide the result by the amount of values. Considering our previous sample, ({11; 15; 17; 17; 18; 19; 21; 21; 21}), the average would be 11 + 15 + 17 + 17 + 18 + 19 + 21 + 21 + 21 = 160 / 9 = 17, 78. Notice that we divided the sum by 9 because 9 is the number of values in the set.
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The "median" of a set of numbers is the "central number", the one that separates the smallest from the largest by dividing the sample in half. We always examine our sample, ({11; 15; 17; 17; 18; 19; 21; 21; 21}), and we realize that
Step 18. it is the median, because it is the central value and there are exactly four numbers below it and four above it. Note that if the sample is made up of an even number of data, then there won't be a single median. In this case, the average of the two median data is calculated.
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Method 2 of 2: Finding Fashion in Special Cases
Step 1. Remember that fashion does not exist in samples made up of data that appears an equal number of times
If the set has values that are repeated with the same frequency, then there is no data more common than the others. For example, a set made up of all different numbers has no fashion. The same happens if all the data is repeated twice, three times, and so on.
If we change our example set and transform it like this: {11; 15; 17; 18; 19; 21}, then we note that each number is written only once and the sample it has no fashion. The same could be said if we had written the sample like this: {11; 11; 15; 15; 17; 17; 18; 18; 19; 19; 21; 21}.
Step 2. Remember that the mode of a non-numeric sample is calculated by the same method
Samples are usually composed of quantitative data, that is, they are numbers. However, you may come across non-numeric sets and in this case the "fashion" is always the data that occurs with the greatest frequency, just like for samples composed of numbers. In these special cases you can always find the fashion, but it may be impossible to calculate a meaningful mean or median.
- Suppose a biology study determined the tree species in a small park. The data of the study are as follows: {Cedar, Alder, Pine, Cedar, Cedar, Cedar, Alder, Alder, Pine, Cedar}. This kind of sample is called nominal, because the data is distinguished only by names. In this case, fashion is Cedar because it appears more often (five times against the three of the alder and two of the pine).
- Note that for the sample under consideration it is impossible to calculate the mean or the median, since the values are not numeric.
Step 3. Remember that for normal distributions the mode, mean and median coincide
As stated above, these three concepts can overlap in some cases. In well-defined specific situations, the density function of the sample forms a perfectly symmetrical curve with a mode (for example in the "bell" Gaussian distribution) and the median, the mean and the mode have the same value. Since the distribution of the function graphs the frequency of each data in the sample, the mode will be exactly in the center of the symmetrical distribution curve, so the highest point on the graph corresponds to the most common data. Considering that the sample is symmetrical, this point also corresponds to the median, the central value that separates the whole in half, and to the mean.
- For example, consider the group {1; 2; 2; 3; 3; 3; 4; 4; 5}. If we draw the corresponding graph, we find a symmetrical curve whose highest point corresponds to y = 3 and x = 3 and the lowest points at the ends will be y = 1 with x = 1 and y = 1 with x = 5. Since 3 is the most common number, it represents fashion. Since the sample's middle number is 3 and has four values to its right and four to its left, it represents also the median. Finally, considering that 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 + 5 = 27/9 = 3, then 3 is also the mean of the whole.
- Symmetrical samples that have more than one fashion are an exception to this rule; since there is only one mean and one median in a group, they cannot coincide with more than one mode simultaneously.
Advice
- You can get more than one fashion.
- If the sample is made up of all different numbers, there is no fashion.
