Measures of Central Tendency : Mean, Median and Mode

Measures of central tendency in statistics  refers to a value which characterizes all the the data.  Mean, median and mode are very important measures of central tendency. Measures of central tendency is very fundamental in data science,  machine learning and data analytics areas. Moreover, if you want to make a carrier in data analytics you have to go through concept of central tendency.

Important- To know about organization of data see the post

Frequency Distribution in Statistics

1- Mean-

Mean is a very frequent used measure of central tendency.  Further, it is used for individual, discrete and continuous series data.

1.1 Mean for Individual Series Data-

Mean for individual series refers to sum of values divided by number of values. If x₁, x₂, x₃,………………., x_n are values then arithmetic mean for individual series is given by.

Mean=Σ(x₁₊x₂₊ x₃,……………….+x_n) / n

For example marks obtained by 10 students in a class 35, 28, 38, 25, 28, 38, 28, 45, 35, 38. The the mean of obtained marks will be.

Mean of Obtained Marks= (35+28+38+25+28+38+28+45+35+38)/10= 33.80

1.2 Mean for Discrete Series Data-

For discrete series mean is calculated as follows.

If x₁, x₂, x₃,………………., x_n are values and their corresponding frequencies are f₁, f₂, f₃,………………., f_n

Then mean is

Mean = Σ(x₁ * f₁) + (x₂ *f₂)+ x₃ *f₃ +……………..+ (x_n *f_n) / N
Where
N= Σ(f₁ + f₂ + f₃ +……………….+ f_n)

In the below frequency table,  marks obtained by students in a class and their occurrences (frequency) are given by.

The mean of marks obtained by students  can be calculated as.

Mean of Marks = (5*1 + 12 *1 + 15 *2 +  25 *3 +…………+ 97 *1)/109 = 53.990825688

Frequency Distribution

1.3 Mean for Continuous Series Data-

To calculate mean for continuous series data you have to follow this basic procedure.

For each class calculate middle value using this formula

M_i = (Lower Limit_i + Upper Limit_i) / 2
For example for row first in table
M₁ = (0 + 10)/ 2 = 5
M₂ = (10 +20) / 2 = 15

Replace x_i with M_i in discrete series mean formula.
i.e
Mean = Σ(M₁ * f₁) + (M₂ *f₂)+ M₃ *f₃ +……………..+ (M_n *f_n) / N
N= Σ(f₁ + f₂ + f₃ +……………….+ f_n)
Then mean for the data contained by table would be
Mean = (5 * 1 + 15 * 3 + 25 * 6 + 35 * 10 + 45 * 20 + 55 * 30 + 65 * 20 + 75 * 10 + 85 * 6 + 95 * 3) / 109
Mean = 54.541284404

2- Median-

In a data set median is middle point  and characterizes whole data set. Furthermore, it can be calculated for individual series data, discrete series data, and continuous series data.

2.1 Median for Individual Series Data

To calculate median of an individual series data you have to see whether number of data odd or even. Then short series data into increasing order.

2.1.1 Median for Individual  Odd Series Data Set

If data list contain odd numbers of  n values  then calculate  (n+1)/2  which will be the position of the median.

For example see the individual series data set

35, 28, 38, 25, 28, 38,  45

Arrange the data into increasing order

25, 28, 28, 35, 38, 38, 45

Number of data are 7 which is odd

Calculate (7+1)/2=4 term   position ( value is 35)

Then median is 35

2.1.1 Median for Individual  Even Series Data Set

If number of data points are even then you have to calculate n/2^th and (n+2/2)^th and take their average which will be median.

Again add one more value  50

Then series will be

25, 28, 28, 35, 38, 38, 45, 50

The number of data points 8, which is  even

Then n/2^th and (n+2)/2^th term will be 35, and 38

Take their average

i.e (35  + 38)/2 = 36.5

Therefore median is 36.5

2.2  Median for Discrete Series Data

See the table below, it contains two columns values and their frequencies. To calculate median for discrete series you need to add another column cumulative frequency.

Frequency Distribution

Then the resultant table you can see below, and  data values are already sorted.

To calculate median calculate N+1/2= 109+1/2 = 55 and corresponding median  value is 50.

If number of data points are even you have  to calculate N/2 and N+2/2 and take average of corresponding values.

2.3 Median for Continuous  Series Data

To calculate median for continuous series data you have to see the formula

Median = L + [(N/2-F)/f] * w

Where L is lower limit

N is sum of frequencies

F is cumulative frequency before median class

f is frequency of median class

w is width of interval

See the table below there are two columns, you have to add another column cumulative frequency.

The resultant table will be

From the above table

You can observe

N=109,  N/2 = 54.5

Then median class is 50-60  and L = 50

You can also observe F= 40 , f=30 and  w=10

Then median

Median =  50 +[ (54.5-40)/30] * 10

Median = 50 + 4.83 = 54.83
The the median is 54.83

3-Mode-

In essence, the value which appears frequent in data set is mode of a data set.

3.1 – Mode of Individual Series Data

Consider the data set 35, 28, 38, 25, 28, 38,  45, 35, 38. You can see that 38 appears 3 times which is most frequent occurrence. Then the mode of the data set is 38.

3.2 Mode of  Discrete Series Data

See the data set

Frequency Distribution

You can observe that frequency of value 50 is highest which is 15. Then mode of discrete data series is 50.

3.3 Mode of Continuous Series Data

For continuous data series you have to use formula which is

Mode = L + (f₁ – f₀)/ (2 * f₁ – f₀ – f₂)]
L is lower limit of the model class
where f₁ is frequency of  the modal class
where f₀ is frequency of the  pre-modal class
where f₂ is frequency of  the post-modal class

See the table you can observe that model class is 50-60.

(Modal class has  the highest frequency)
And f₁= 30
f₀ = 20
f₂ = 20
L=50
Then
Mode = 50 +[(30-20)/ (2 * 30 – 20-20)] = 50 + 0.50 =50.50

Then the mode of continuous data series will be 50.50.

Conclusion-

To sum up, in this post I have explained about measurement of central tendency in statistics. Which is the first step towards learning statistics. I addition I have tried my best to make you understand hope it will be helpful for you.

References-

1. Wilcox, R.R. and Keselman, H.J., 2003. Modern robust data analysis methods: measures of central tendency. Psychological methods, 8(3), p.254.
2. Balota, D.A. and Spieler, D.H., 1999. Word frequency, repetition, and lexicality effects in word recognition tasks: Beyond measures of central tendency. Journal of Experimental Psychology: General, 128(1), p.32.
3. Smothers, C.D., Sun, F. and Dayton, A.D., 1999. Comparison of arithmetic and geometric means as measures of a central tendency in cattle nematode populations. Veterinary Parasitology, 81(3), pp.211-224.
4. Web link Retrieved on 29.09.2019, Mean, median, and mode review ,  https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/mean-median-basics/a/mean-median-and-mode-review.
5.   Web link Retrieved on 29.09.2019,  How to Calculate Median in 2 different Series? Explained! http://www.economicsdiscussion.net/central-tendency/how-to-calculate-median-in-2-different-series-explained/2465.
6.  Web link Retrieved on 29.09.2019, Calculation of Mean, Median and Mode, https://www.toppr.com/guides/economics/measures-of-central-tendency/calculation-of-mean-median-and-mode/