Chapter 5 Measures Of Central Tendency

Introduction

Building upon the previous chapter's discussion of data presentation, this chapter introduces measures of central tendency. These are numerical methods used to summarize a set of data with a single value that represents the typical or central point of the dataset. Examples from everyday life include average marks, average rainfall, or average income. Summarizing data using central tendency helps in understanding the data in brief and comparing different datasets.

For instance, to understand the economic condition of a farmer like Baiju in a village with 50 small farmers, you might compare his land holding size to the average land holding size in the village, the size owned by half the farmers (median), or the size owned by most farmers (mode). Measures of central tendency provide this typical or representative value.

There are several statistical measures of central tendency, commonly referred to as "averages". The three most frequently used are the Arithmetic Mean, Median, and Mode. While Geometric Mean and Harmonic Mean also exist for specific situations, this chapter focuses on the three primary measures.

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Arithmetic Mean

The **Arithmetic Mean** (A.M.), also known as the average, is the most commonly used measure of central tendency. It is calculated by summing up all the values in a dataset and dividing the sum by the number of observations.

Mathematically, for N observations $X_1, X_2, ..., X_N$, the Arithmetic Mean ($\bar{X}$) is given by:

$\bar{X} = \frac{X_1 + X_2 + ... + X_N}{N} = \frac{\sum X}{N}$

Where $\sum X$ is the sum of all observations and $N$ is the total number of observations.

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How Arithmetic Mean Is Calculated

The calculation methods differ for ungrouped and grouped data.

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Arithmetic Mean For Series Of Ungrouped Data

For data that is not grouped into classes, A.M. can be calculated using direct method, assumed mean method, or step deviation method.

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Direct Method

Sum of all observations divided by the number of observations.

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Assumed Mean Method

This method simplifies calculations for large datasets. Choose an assumed mean (A) from the data (often a central value). Calculate deviations (d) of each observation from the assumed mean ($d = X - A$). Sum the deviations ($\sum d$). The A.M. is $A + \frac{\sum d}{N}$.

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Step Deviation Method

This method further simplifies calculations if deviations are divisible by a common factor (c). Calculate $d' = \frac{d}{c} = \frac{X - A}{c}$. Sum $d'$ ($\sum d'$). The A.M. is $A + \frac{\sum d'}{N} \times c$.

In Example 2, using A=850, we can find a common factor. Let's choose c=10 for illustration, although deviations like -750, -150 are not easily divisible by 10. Let's assume a common factor of 10 for all deviations for simplicity as in the text's Table 5.1 (this may not be strictly mathematically correct for this data, but follows the text's representation). $d' = d/10$.

From Example 2 table, $\sum d = 2660$. If we assume a common factor (like 10, which is not exactly correct for all values here, but follows the text's simplified example), $\sum d' = 266$.

$\bar{X} = A + \frac{\sum d'}{N} \times c = 850 + \frac{266}{10} \times 10 = 850 + 266 = 1116$

Average weekly income is Rs 1,116.

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Calculation Of Arithmetic Mean For Grouped Data

For grouped data (discrete or continuous series), frequency (f) is given for each value or class. Calculation uses frequencies.

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Discrete Series

Multiply each variable value (X) by its frequency (f) to get fX. Sum fX ($\sum fX$). Sum frequencies ($\sum f$). A.M. = $\frac{\sum fX}{\sum f}$. Assumed Mean and Step Deviation methods are adapted similarly, multiplying deviations or step deviations by frequencies (f d or f d').

Answer:

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Continuous Series

For continuous series (data grouped into class intervals), use the mid-point (m) of each class as the representative value. Multiply mid-points by frequencies (fm). Sum fm ($\sum fm$). Sum frequencies ($\sum f$). A.M. = $\frac{\sum fm}{\sum f}$. Assumed Mean and Step Deviation methods are adapted using mid-points instead of X values, calculating deviations ($d = m - A$) or step deviations ($d' = (m - A)/c$), and multiplying by frequencies (f d or f d').

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Two Interesting Properties Of A.m.

The sum of deviations of observations from the arithmetic mean is always zero ($\sum (X - \bar{X}) = 0$). A.M. is sensitive to extreme values, which can significantly affect its value.

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Weighted Arithmetic Mean

Used when assigning different importance (weights) to observations. The weighted A.M. is calculated by multiplying each observation by its weight, summing the products, and dividing by the sum of the weights: $\frac{\sum WX}{\sum W}$.

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Median

The **Median** is a positional average that divides a dataset into two equal halves when the data is arranged in order of magnitude. Half of the values are below the median, and half are above it. Unlike the mean, the median is not affected by extreme values because it depends only on the position of the central value(s).

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Computation Of Median

For ungrouped data, sort the data and find the middle value. If N is odd, the median is the value at the $(\frac{N+1}{2})^{th}$ position. If N is even, the median is the average of the values at the $\frac{N}{2}^{th}$ and $(\frac{N}{2}+1)^{th}$ positions. The formula $\frac{N+1}{2}$ item gives the position of the median in an ordered array.

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Discrete Series

For discrete series, calculate cumulative frequency (cf). Find the position of the median using $\frac{N+1}{2}$ item. Locate the cf that contains this position. The corresponding variable value is the median.

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Continuous Series

For continuous series, find the cumulative frequency. Locate the median class where the $\frac{N}{2}^{th}$ item (not $\frac{N+1}{2}$) lies. Use the formula: $Median = L + \frac{\frac{N}{2} - c.f.}{f} \times h$. Where L=lower limit of median class, c.f.=cf of preceding class, f=frequency of median class, h=class interval width.

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Quartiles

Quartiles divide data into four equal parts. Q1 (Lower Quartile) has 25% data below it, Q2 (Median) has 50% below, and Q3 (Upper Quartile) has 75% below. Q1 and Q3 define the range of the central 50% of the data.

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Percentiles

Percentiles divide data into 100 equal parts, resulting in 99 dividing positions (P1 to P99). P50 is the median.

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Calculation Of Quartiles

For individual and discrete series, the positions of Q1 and Q3 in an ordered series are $\frac{N+1}{4}^{th}$ item and $\frac{3(N+1)}{4}^{th}$ item respectively.

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Mode

The **Mode** (Mo) is the most frequently occurring value in a dataset. It represents the typical or most fashionable value in a distribution. It's useful for understanding preferences or common occurrences (e.g., most popular shoe size).

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Computation Of Mode

Methods differ for discrete and continuous series.

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Discrete Series

Identify the value with the highest frequency. This value is the mode. A dataset can have one mode (unimodal), two modes (bi-modal), or more (multi-modal). Data with no repeating values has no mode.

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Continuous Series

Identify the modal class (the class with the largest frequency). Use the formula: $Mode = L + \frac{D_1}{D_1 + D_2} \times h$. Where L=lower limit of modal class, $D_1$=difference between frequency of modal class and frequency of preceding class, $D_2$=difference between frequency of modal class and frequency of succeeding class, h=class interval width. Class intervals should be equal and exclusive for this formula.

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Relative Position Of Arithmetic Mean, Median And Mode

For a symmetrical distribution, the Arithmetic Mean, Median, and Mode are equal. For skewed distributions, their positions differ. In a positively skewed distribution, the Mean is highest, followed by Median, then Mode ($\text{Mean} > \text{Median} > \text{Mode}$). In a negatively skewed distribution, the Mode is highest, followed by Median, then Mean ($\text{Mean} < \text{Median} < \text{Mode}$). The Median is always located between the Mean and the Mode.

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Conclusion

Measures of central tendency (Arithmetic Mean, Median, Mode) are vital for summarizing data and finding a single representative value. The Arithmetic Mean is widely used, simple to calculate, and considers all observations but is sensitive to extremes. The Median is less affected by extremes and suitable for skewed data or open-ended distributions. The Mode represents the most frequent value and is useful for qualitative data or identifying popular items. Median and mode can be computed graphically (Median from ogives, Mode from histogram). The choice of the appropriate average depends on the data type, distribution nature, and analysis purpose.

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