Q 224. If the data is not expected to be normally distributed, median is used to indicate performance. Provide as many diverse examples as you can to indicate metrics where median is commonly used.

 

 

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Mean (Average)

Mean  is the best measure of central tendency in normally distributed data without significant outliers. As  large number of  distributions are symmetrical, mean  represents the  true estimate of distribution. Example: Mean height, weight etc.

 

Mode

Mode is  the most repeated value in a set of data. Like,  people become more inclined to things, that are undertaken by majority of the people. 

Median

The median is the  mid value that divides  a set of values into top and bottom 50%.  

 The income distribution in a country  is asymmetrical, with 20% of population, accounting for  major proportion of wealth in the country and remaining 80% of the people  have lower income, in the way that wealth of top 20% is equal to bottom 80%.

In this case, mean income will give a false and biased picture, due to distribution peaks in two different regions. Median will  be the best representer of  the  income of the people in the country.

 

In India, many educational institutes place their advertisements to attract students by stating the ''placement packages'' of their passing students either due to on campus selection attract students using “placement packages”.  In this example, average placement package, which is commonly quoted, is a wrong way of assessing the students. Rather, median serves as the best measure, as the salary range is quite wide, for example  for those selected for  India location -20 students(salary up to 3 million INR)  and  those  for USA location-5 students (Salary in range of 8 to 10 million INR after conversion  of USD to INR)

 

Median also finds use in measurement of commonly measured  health indices such as  blood pressure. If we measure the blood pressure of 5000 persons in a community health survey and tabulate the  systolic and diastolic pressures separately, mean will give an  erroneous  impression,  as 10-15%. of the patients may have very  high systolic and diastolic blood pressures, much above the normal reference range( say, Systolic> 200mm of Hg and Diastolic> 140mm of Hg , which is not represented by majority) . Here, median will be the best measure for blood pressure levels of the community people and can be used to initiate  health intervention for the community.

 

 

 

Median: - Median is a measure of central tendency. We need to arrange the data in order from smallest to largest value for calculating the median, if it is odd number of data then middle value of data would be a median and if there is an even number of data then median is the average of the two middle values. We use the median for measuring the central tendency if data have outliers.

For example – if we have data set of (3,13,2,34,11,26,47) and we need to calculate the median then we arrange the data sent from smallest to largest value –(2,3,11,13,26,34,47) . it is an odd number of data set hence middle value of data 13 is a median of this data set.

If we have even number of data set like – (2,3,11,13,17,26,34,47) then median is average of two middle of 2 value (13+17)/2= 15

Uses of Median as a central tendency:

Type of data	Best measure of center tendency
Ordinal data	Median
Interval /ratio data (skewed)	Median

 

We use median as a central tendency in service sector like customer feedback (ordinal data), NPS (ordinal data), customer age group visited in malls, height of visitor in water park. we use median where data is ordinal or skewed

The median is used as an indicator of central tendency when there is high skewness or asymmetry in the underlying data. Comparison to the mean, which is dynamic, the median is completely static

Typical examples of datasets where the median is preferred are include salaries, real estate prices, etc. 

Having the above, it is observed that the median is quite easily jumped upon as a preferred method of evaluation, especially when data is found to be not-normal, considering that it is more robust and makes one use the distantly related concepts of non-parametric tests. I would like to take this opportunity to mention a few points. 

As the ANOVA, and other variance/ mean “parametric” evaluation techniques were developed in the early parts of the century, there were a few basic assumptions which were considered important: 

1.    Observations in a data set need to be independent of each other, that is, the assumption is that the fact that you have observed a value in one group has no effect on the likelihood of observing another value in either group
2.    Data sets evaluated need to have equal variances
3.    The assumption of Normality

Belief in the assumptions was sacrosanct, and if you have data sets which, for example, do not have equal variances, you simply, at the time, did not proceed to evaluate an F-Test

However, the above assumptions were difficult to “test”, basically owing to limited computing power in the early parts of the century. As time passed, in the 1950-70s, there were these “robustness” studies performed, which gave good light to clarify the above

1.    The independence factor is extremely important, and having pairs of data makes them correlate and covary
2.    As long as data sets have equal (or nearly equal) sample sizes, the equal variance assumption can be forgiven
3.    Normality: this assumption can safely be ignored in practice, and these robustness studies gave good confirmation that non-normality of data, (especially for the t-distribution), has trivial effect on the results. These studies did however iterate that it is important for Residuals obtained, after testing to be normally distributed, rather than the data itself. 

Not to mention, the median and it’s accompanies tests do have good applications when the critical assumptions above are not being met. 
So let's all think again before jumping on the Median !!

Benchmark Six Sigma Expert View by Venugopal R

Using median as a measure of central tendency helps to avoid effect of outliers. For those who need a clarity on fundamental behavior of mean and median, the following simple example will help.

 

Consider as set of nine data points representing the minimum time in days between failures for nine similar equipment.

70, 248, 2400, 240, 2, 1460, 230, 180, 440

The mean for the above data is 586 whereas the median is 240.

 

Now consider the data set below, which is same as above except that the maximum value has further increased from 2400 to 4800

70, 248, 4800, 240, 2, 1460, 230, 180, 440

The mean has shot up to 852, whereas the median remains unaffected at 240.

 

In the above situation, the median is a more realistic representation as a measure of central tendency of the data.

 

Few examples where the median may be a better choice:

 

1.    Income data in an organization: It is quite possible that there could be a few high paid individuals, by which the mean could be severely biased, hence median is preferable.

 

2.    Age of employees in a society: A few very senior citizens among a majority of people being in the lower middle age band, could give a non-normal distribution.

 

3.    Customer satisfaction surveys using a Likert scale of 1 to 10: A very few customers voting on the upper or lower extreme could distort the reality – hence usage of median helps.

 

4.    Life expectancy based on a specialized treatment: For instance if most patients had a post treatment life span in the range of 10 to 15, one odd patient living for 45 years could provide an unrealistic expectancy, unless we use median as a measure of performance.

 

5.   The comparative tests performed on non-normal distributions, knows as non-parametric tests are based on usage of median. Examples of such tests are 1-Sample sign, Wilcoxon Signed rank, Mann Whitney, Kruskal Wallis, Moods Median.

Use of median for examination evaluation. 

Use of median to calculate country real GDP. 

 

Well median can be used effectively in preparation of organizational matrix where there are huge diaparities. One practical example can be in the case of preparation of salary bands. If in one designation category there is a huge disparity in salaries and the median can be used to form salary bands for a particular category/ designation of employees more effectively.

One example is while taking cycle tym reading of an mechanist.

Median is a statistical tool for data representation which is a measure for central tendency. In cases where, Mean which is the average of all the data, are not able to represent the correct picture of data requires Median as an indicator which is the central value in the entire data sets and make a clear distinction between first half and second half of the data sets.

 

EXAMPLE 01 : The sales performance for 9 units of an apparel industry showed above average performance for last 3 quarters. So If Team goes by Mean as an indicator for statistical performance monitoring, management would assume that, the KPI is performing well. However if the team distributes the data for all 9 units using Median as an indicator, it was observed that 4 of the 5 units are performing very low in the sale performance, and the other 4 are performing exceedingly well. That kind of statistical representation gives a clear picture to the management that, those 4 units which are not performing well need to be prioritized and given special attention for improvement. But this kind of analysis would not have been possible, if merely we could have used Mean as an indicator.

 

EXAMPLE 02 : One of the pharmaceutical manufacturing organisation has got 17 assembly lines. The management is worried about the Overall Equipment Effectiveness (OEE) performance of the Assembly line as a whole. Against the target OEE of 60%, consider if the average,i.e., Mean OEE of assembly lines come out for past one year comes out to be 63%, than obviously anyone can make out that, we are performing well in terms of average OEE performance of the site. However, if we would like to go into detail into details for bringing continuous improvement in the system, we will be extrapolating the data in terms of median representation. That will give us the clear picture in terms of Which assembly lines among 9 lines are giving poor performance in OEE against the central value and which of them are performing well. This will be a significant input for the management to focus on the assembly lines which are constraints/ poor performer. Hence, Median can be a good tool for ANALYZE phase of Problem solving.

 

EXAMPLE 03 : Lets take another case from a service industry. Suppose there are 121 delivery boys for the Food delivery startup. Operational excellence team started to monitor the data for delivery time accuracy for each of the delivery boys in its dashboard. Even in this case, like above examples, Median can be a good tool to understand the best performers and least performer against the central value. 

 

Thus, being the med point in the entire data sets, Median helps in structural distribution of data, which brings more clarity to the problem solving team, on which area to be focused for improvement as a priority.

Having provided three good examples, Dr. Shashikant Adlakha is the winner today. Please have a look at Expert View (Venugopal's respose) too.