Measures of central tendency assignment help

In the broad field of statistics, a typical value attached to probability distribution is called central tendency or sometimes measure of Central location. In general they are often called averages since they are descriptive measures that guides us to where the variable centre lies in a sample collected for the purpose of an observation. We need to keep in mind that the central tendency of a distribution differs from dispersion and variability which arises out of characterizations of properties of distributions.

Colloquially, arithmetic mean, median and mode are the most common measures. Where mode can be used for both quantitative and qualitative data, mean and median can only be use on quantitative data.

The Mode

The formal definition of Mode goes by:

“Obtain the frequency of each observed value of the variable in a data and note the greatest frequency.”

Some conditions that come with measuring mode are as follows: a) the mode of the variable does not exist if the maximum frequency is one (meaning that no other value occur more than one time). B) The mode measured for the variable refers to the greatest frequency of a value of two or greater.

In simple terms, the mode of a data set is the value of that variable that occurs the most.

A mode can be one number or more. A frequency table needs to be constructed to measure the mode. By using classes based on a single value, the process of determining the mode becomes an easy process. 

As an example, we will look at a frequency distribution table 40 people with their blood types.

Looking at the frequency table, we can determine that the sample mode of blood types in A. Further using a statistical technical tool called SPSS, we analyse the descriptors and find the needed results. 

A continuous variable or in some cases discrete variables having multiples values can have distinct measurements such as height or weight. This will yield to absolutely no mode since every observed value has one frequency. This is when the collected data is grouped into class intervals on the basis on which the frequencies can be calculated. This helps us solve the problem of obtaining a mode. The intervals of the class with the maximum frequency yields the mode. 

The Median

The formal definition of The Median goes by : 

“Arrange the observed values of variable in a data in increasing order. If the number of observation is odd, then the sample median is the observed value exactly in the middle of the ordered list but if the number of observation is even, then the sample median is the number halfway between the two middle observed values in the ordered list.”

In simple terms, the median of a given quantitative variable, can be considered as that value which divides the sample in half so as to give a condition where one half is equals or less than the median and the second half of bigger than or exactly the same of the median value. To obtain this, we need to first arrange the  data collected in ascending order, after which we’ll determine the middle vale of the list. This is called the median. In the other case where the total observations are even in number.  We calculate is by adding the middle values and dividing it by two.

Mean

The formal definition of mean goes by :

“The sample mean of the variable is the sum of observed values in a data divided by the number of observations.”

As mentioned above, the most popular measure if central tendencies for a quantitative variable is called the Mean. It is often referred to as taking average by most people. 

For the ease and efficacy of providing the example, we can represent the assumed variable by taking a symbol instead of a number to avoid it from anchoring to the same set of numbers. Here the variable is denoted for the symbol xi denoting the ith observation, the sample size can be denoted by the symbol n. 

The mean can be calculated as (x1 + x2 + x3 + · · · + xn)/ n . To ease the formula even further for the better and efficient use during research, The sigma sign in the Greek letter is used to denote the mean. x1 + x2 + x3 + · · · + xn  is the sum which will be annotated to Xn i=1 xi , and can be read as the addition of xi from number 1 to n. In words this will mean that the mean sample is given by the addition of the value depicted by symbols x-1, x-2, x-3, .  xn in a data set and then divided by the total number of observations n. 

The question that arises now is that which measure to choose and when.

According to statistical theory. The mode is the best form of central tendency measurement and should be used to calculate the centre of a qualitative variable in a data set. If a symmetric distribution is portrayed and the variable seem to be quantitative, mean as a measurement of central tendency is ideal. However, in case of skewed or biased distribution table, the quantitative variable should be measured by using Median. This can also be due to the fact that value of the mean can be highly skewed if an observation falls far off. This value which falls far far away from the data set left can skew the entire sample and is called an outlier. Since the median, mode and mean of the entire sample are in correspondence with their measurements of the population mean, mode and median which are traditionally unknown. These unknown population median, mode and mean can be estimated with the help of the sample mean, sample median and sample mode.