There are many types of averages in statistical mathematics. However, out of them, the three most common and widely used are mean, mode and median. Knowing about these three is extremely important for any student as it is a part of almost every statistical course. Therefore, having a detailed knowledge of each of them is extremely necessary to have a better understanding of range math. Mean has many definitions in mathematics, all of which depend on the context of study. However in statistics, the mean is an expected value used to measure the central tendency of either a random variable or a probability distribution. The distribution of mean for a variable Y are a set of discrete values and the mean for Y is calculated by taking the sum total of all the possible values and is divided by the total number of values. This gives the value of the central tendency. In range math, the mean may sometimes be confused with the mid-range or the mode and median. However, mean is none of these, but an arithmetic average of all the given values, that is, the mean = sum of all the discrete values/ total number of discrete values.
For example, let us calculate the mean for the set of numbers 13, 15, 16, 12, 13, 18, 7. The total number of values is 7 for the given example. The mean here is nothing, but the average of all the values. For this reason, it is also called the arithmetic mean. It is also denoted by the subscript AM. The mean for this value is calculated as follows
Mean= 13 +15+16+12+13+18+7/7;
In mathematics, there are many types of mean, however, the arithmetic mean which has been discussed above is a part of range math. Apart from this, the other most commonly used mean is the geometric mean. It is also denoted by the subscript GM. Now, the geometric mean uses the product of the numbers instead of their sum and can be calculated for positive numbers only as it deals with under root operation which may give imaginary results for a negative value. This method is most commonly used in calculating the time of radioactive decay, the half-life of elements and the growth of a population, among others. It is determined by calculating the product of the number which are raised to the power of the inverse of the nth root, where n is equal to the number of terms. For example, to calculate the geometric mean of the numbers 36, 50, 45, 70 and 4. The product is first calculated which comes out to be 24300000. This is raised to the power of 1/5 as there are total 5 terms. The geometric mean finally comes out to be 30.
Another most common type of mean calculation that is widely used in range math is the harmonic mean. It is extremely useful in calculating speed or to find relations of units that are defined by some number sets. It is also denoted by the subscript HM. Calculating harmonic mean is very simple and can be illustrated with a simple example. For the set 36, 50, 45, 70 and 4 where the number of terms is 5, the harmonic mean is determined by first calculating the sum of the inverse of these numbers and then dividing 5 by this sum. So the sum of the numbers 1/36 +1/50 + 1/45 +1/70+ 1/4 will come out to be 1/3 and when 5 is divided by it, that is, 5/ (1/3); we get the answer as 15. Also, it has been found that the value of the arithmetic mean is always greater than the geometric mean which in turn is greater than the harmonic mean. The equality between the three holds only when the elements of the sample taken are equal in all the three. These three types of mean have a wide variety of applications in range math.
So, AM> = GM> = HM;
Median has very useful applications in range math as well as in statistical and probability based maths. It is basically a number that separates the higher half of the sample space from the lower half. However for this, it is important that the numbers in the sample data are arranged from the lowest number to the highest. In fact, this is the first step that needs to be followed while finding out the medium for a given data sample. A medium can only be defined for data which is one dimensional and ordered. It also does not take into account the distance metric. Calculating median is a simple process which involves a systematic approach. As mentioned above, the first step will be to arrange them from the lowest to the highest value, that is, in an increasing order. After that, the value of n has to be determined which is equal to the number of observations in a given sample space. Calculating the value of n will tell us whether the sample data is even or odd. In each case, a different formula is employed to find the median.
In case the sample space is even, then the formula for median is the sum total of the (n/2)th term and (n/2+1) th term divided by 2. For example, to find the median for a set of numbers 1,2,7,6,2,8 it is important to arrange them in the correct order, that is, 1,2,2,6,7 and 8. Where n is equal to 6. Thus, by using the formula we can find out the (6/2) th term which is 2 and (6/2 +1) th term which is 6. Now adding the two will give us 8 and finally dividing it by 2 will give us the median, which is 4 in our case.
Now for an odd sample space, say, 1, 2,8,5,7. We again need to follow the same step of arranging the terms in ascending order and thus, we get 1,2,5,7 and 8 and thus, the value of n is 5. The formula to calculate the median for odd sample space is the value of the ((n+1)/2) th term which in this case will be the value of ((5+1)/2) th term. This comes out to be the 3rd term whose value is 5. Thus, the median for the given sample data is 5. Apart from in range math, median has many applications. It is particularly useful in image processing where the image is corrupted by salt and pepper noise. In this type of noise, each pixel either turns black in colour or changes to deep white thereby, increasing the image contrast and noise. To get rid of such noise effects, median filters are used which make use of 3X3 squares in the neighbourhood of the pixel to prevent degradation of the image.
Mode is a range math value and is defined as that number in the sample space which occurs most often or has the most frequency of occurrence in that sample data. In fact, we can define it as the value that has to be sampled the most frequently. For example, in the data set 1,2,3,3,2,2,6,2,2,7; the mode of this data set will be 2 as this is the number that appears for the maximum number of times in a mode. However, for some sets, more than one mode may exist. Like 1,3,3,5,6,2,2 is a bimodal set which has two values for the mode. Similarly, a multimodal set has more than 2 or more modes.
Unlike the concept of median and mean, the formula of mode can be applied on a nominal data. We can also find the mode for a continuous distribution where no two values are same. We first need to discretize all the values in the data to make a histogram by assigning a frequency value of equal space intervals and then replacing these values with the midpoints of those particular intervals they have been assigned. The maximum or the peak value reached in the histogram gives the value of mode. In the case of a mid-sized or a small-sized sample, the sensitivity of the outcome is based on the interval width and the interval must not be either too wide or too small. In fact, it is best to have all the values concentrated in a sizeable fraction of relatively small intervals of nearly 5-10.
Calculating mean, median and mode is extremely simple if the following steps are followed systematically. They are part of range math that finds many applications in our day to day life.