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What is Measurement of Central Tendency?
Measurement of Central Tendency refers to the statistical methods used to identify a single value that best represents a set of data by indicating the center of the distribution. It summarizes a dataset with one representative value around which all other values cluster. This central value is known as an "average" or "central value" and is essential for comparing, analyzing, and making decisions based on data.
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Objectives of Measurement of Central Tendency
The main objectives of measuring central tendency include:
1. To summarize large data into a single representative value.
2. To make comparisons between different datasets.
3. To aid in decision-making by simplifying complex data.
4. To identify the trend or behavior of a data set.
5. To provide a base for other statistical analyses like dispersion, correlation, and regression.
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Requirements of an Ideal Measure of Central Tendency
An ideal average or central tendency measure should:
1. Be clearly defined and easy to understand.
2. Be based on all observations in the data set.
3. Be simple to compute and not influenced by extreme values.
4. Be capable of further mathematical treatment.
5. Be stable and consistent in repeated samples.
6. Be not unduly affected by sampling fluctuations.
7. Be representative of the entire dataset.
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Types of Statistical Averages (Measures of Central Tendency)
There are three main types of statistical averages:
1. Arithmetic Mean (AM)
2. Median
3. Mode
Other lesser-used averages include:
Geometric Mean (GM)
Harmonic Mean (HM)
Quadratic Mean or Root Mean Square
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Arithmetic Mean (Arithmetical Average)
The Arithmetic Mean is the sum of all values divided by the number of values. It is the most commonly used average.
\text{Arithmetic Mean (AM)} = \frac{\sum X}{n}
Methods of Calculating Arithmetic Mean
1. Individual Series:
AM = \frac{\sum X}{n}
AM = \frac{\sum fX}{\sum f}
3. Grouped Frequency Series:
Direct Method
AM = \frac{\sum fX}{\sum f}
AM = A + \frac{\sum fd}{\sum f}
Step-Deviation Method
AM = A + \left(\frac{\sum fd'}{\sum f}\right) \times h
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Median – Meaning and Computation
Median is the middle value of a dataset when arranged in ascending or descending order. It divides the data into two equal halves.
Calculation of Median
1. Individual Series: Arrange data and find the middle term.
If n is odd: Median = Middle value.
If n is even: Median = Average of two middle values.
2. Discrete Series: Use cumulative frequency to locate the median class.
3. Grouped Data:
\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f}\right) \times h
L = lower boundary of median class
N = total frequency
F = cumulative frequency before median class
f = frequency of median class
h = class width
Merits of Median
Not affected by extreme values
Can be found graphically
Useful for open-ended classes
Demerits of Median
Not based on all observations
Not suitable for further algebraic treatment
Less precise in grouped data
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Mode – Meaning, Definition, and Computation
Mode is the value that occurs most frequently in a dataset.
Grouped Data Mode Formula
\text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h
L = lower boundary of modal class
f₁ = frequency of modal class
f₀ = frequency before modal class
f₂ = frequency after modal class
h = class width
Merits of Mode
Easiest to compute
Not affected by extreme values
Useful in qualitative data (e.g., fashion size)
Demerits of Mode
May not exist or may be multiple
Not suitable for further mathematical operations
Less reliable with small or irregular datasets
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Merits, Demerits, and Applications of Various Averages
Measure Merits Demerits Application
Mean Simple, uses all data, good for further analysis Affected by outliers Salary analysis, scientific research
Median Not affected by extreme values, easy to understand Ignores most data points Income distribution, real estate
Mode Easy for qualitative data, useful in large samples Can be ill-defined or multiple Consumer behavior, fashion, retail
Geometric Mean Accurate for growth rates Not defined for negative/zero values Population, interest rates
Harmonic Mean Ideal for rates (speed, prices) Complicated to calculate Time, travel, cost analysis
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Conclusion
The measures of central tendency—mean, median, and mode—are fundamental tools in statistics to represent data using a single value. Each has its strengths and limitations, and the choice of measure depends on the nature of the data and the specific context in which it is applied. Mastery of these concepts is essential for meaningful data interpretation in fields like economics, business, psychology, and public policy.
