Measures of Central Tendencies, otherwise known as Measures of Location or Averages, are statistical tools used to summarize a data set by identifying the central point within a distribution. They give a single value that represents a group of values. Generally, measures of central tendency are used for data description and comparison.
The three major measures of central tendency are:
Mean (Arithmetic, Geometric & Harmonic), Median and Mode.
(A) MEAN
Arithmetic Mean (A.M.)
The mean is the arithmetic average of a set of numbers. It is computed by adding all the numbers in a data set and dividing by the number of observations (Direct Method). Alternatively, it can be obtained using the Short-Cut (Assumed Mean) Method.
i. Direct Method for Ungrouped Data
<\bar{X}=\frac{<\sum X_i}{n}>
or
[
\bar{X} = \frac{X_1 + X_2 + X_3 + \cdots + X_n}{n}
]
ii. Direct Method for Grouped Data
*Class Midpoint (x)
[
x = \frac{\text{Lower class limit} + \text{Upper class limit}}{2}
]
**Example:
[
x = \frac{11 + 12}{2} = \frac{23}{2} = 11.5
]
For grouped data:
[
\bar{X} = \frac{\sum fx}{\sum f}
]
iii. Assumed Mean Method
For Ungrouped Data**
[
\bar{X}_{A.M.} = A + \frac{\sum d}{n}, \quad \text{where } d = X - A
]
For Grouped Data**
[
\bar{X}_{A.M.} = A + \frac{\sum fd}{\sum f}
]
Example Table**
| Class Interval | f | X | d |
| :------------: | :-: | :-: | :-: |
| 15 – 17 | 6 | 16 | |
| 18 – 20 | 7 | 19 | |
| 21 – 23 | 6 | 22 | |
| 24 – 26 | 10 | 25 | |
(B) Geometric Mean (G.M.)
[
\text{G.M.} = (X_1 \times X_2 \times X_3 \times \cdots \times X_n)^{1/n}
]
or using logarithms:
[
\text{G.M.} = \text{Antilog}\left( \frac{\sum \log X}{n} \right)
]
(C) Harmonic Mean (H.M.)
[
\text{H.M.} = \frac{n}{\sum \frac{1}{X_i}}
]
Relationship Among the Means
[
\text{Arithmetic Mean} \geq \text{Geometric Mean} \geq \text{Harmonic Mean}
]
Equality holds only when all observations are identical.
Advantages and Disadvantages of the Arithmetic Mean
Advantages**
* Simple to calculate and interpret
* Uses all data values, providing a comprehensive summary
Disadvantages**
* Sensitive to outliers, which can distort the average
The Median
The median is the middle value in a data set. It splits the data into two equal halves such that an equal number of observations lie on either side.
(A) Median for Ungrouped Data
**Steps:
1. Arrange the data in ascending or descending order.
2. Check whether the number of observations is even or odd.
If n is odd:
[
\text{Median} = \left(\frac{n+1}{2}\right)\text{th item}
]
If n is even:
[
\text{Median} = \frac{\left(\frac{n}{2}\right)\text{th item} + \left(\frac{n}{2} + 1\right)\text{th item}}{2}
]
(B) Median for Grouped Data
[
\text{Median} = L + \left( \frac{\frac{n}{2} - \sum f_b}{f_m} \right)C
]
Where:
* ( \frac{n}{2} ) = median position
* **L** = lower class limit of the median class
* ( \sum f_b ) = cumulative frequency before the median class
* ( f_m ) = frequency of the median class
* **C** = class width
Advantages of the Median
* Robust to extreme values (insensitive to outliers)
* Gives a better central value for skewed data
Disadvantages
* Does not use all data values
---
(C) MODE
The mode is the value that occurs most frequently in a data set.
A data set may be:
* **Unimodal (one mode)
* **Bimodal (two modes)
* **Trimodal (three modes)
* **Multimodal (more than three modes)
* **No mode (if no value repeats)
Conclusion
Measures of central tendency—Mean, Median, and Mode—provide essential summaries of data by identifying its central point. Each measure has unique strengths: the mean uses all values, the median resists outliers, and the mode shows the most frequent value. Understanding when and how to apply these measures ensures accurate and meaningful statistical interpretation.
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