Introduction to Measures of Central Tendency

Once data is collected and organised, the next step is often to summarise the entire set of data with a single, representative number. This is the purpose of a measure of central tendency, also known as an "average". These measures provide a numerical summary that helps explain the data in brief.

We use averages in our day-to-day life to understand and describe large sets of data, such as:

• Average marks obtained by students in a class.
• Average rainfall in a region.
• Average income of people in a locality.

To understand the economic condition of a farmer named Baiju in the village of Balapur, we would need to compare his land holding with that of the other 50 farmers in the village. This requires summarising the data on all land holdings into a single, typical value. A measure of central tendency provides such a value.

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The Three Most Common Averages

There are several statistical measures of central tendency, but the three most commonly used are:

1. Arithmetic Mean: The "average" in the ordinary sense, which considers all values in the dataset.
2. Median: The middle value that divides the dataset into two equal halves.
3. Mode: The value that occurs most frequently in the dataset.

The choice of which average to use depends on the nature of the data and the purpose of the analysis.

Arithmetic Mean

The Arithmetic Mean is the most commonly used measure of central tendency. It is defined as the sum of the values of all observations divided by the number of observations. It is usually denoted by $\bar{X}$.

Formula for Arithmetic Mean

If there are N observations $X_1, X_2, X_3, \dots, X_N$, the Arithmetic Mean is given by:

$ \bar{X} = \frac{X_1 + X_2 + X_3 + \dots + X_N}{N} $

This can be written in a simpler form using the summation symbol ($\Sigma$):

$ \bar{X} = \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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Calculation of Arithmetic Mean

The method for calculating the arithmetic mean varies slightly depending on whether the data is ungrouped or grouped.

1. Arithmetic Mean for Ungrouped Data

• Direct Method: This is the simplest method, involving summing all observations and dividing by the total number of observations.

  Example 1. Calculate the arithmetic mean of the marks: 40, 50, 55, 78, 58.

  Answer:

  $ \bar{X} = \frac{40 + 50 + 55 + 78 + 58}{5} = \frac{281}{5} = 56.2 $

  The average mark is 56.2.

• Assumed Mean Method: When the number of observations or the figures are large, this method simplifies calculations. An "assumed mean" (A) is chosen, and deviations (d) of each observation from this mean are calculated. The formula is:

  $ \bar{X} = A + \frac{\sum d}{N} $, where $d = X - A$

• Step Deviation Method: To further simplify calculations with large numbers, deviations are divided by a common factor (c). The formula is:

  $ \bar{X} = A + \frac{\sum d'}{N} \times c $, where $d' = \frac{X - A}{c}$

2. Arithmetic Mean for Grouped Data

For grouped data (both discrete and continuous series), the frequency (f) of each observation or class is considered.

• Direct Method (Discrete Series):

  $ \bar{X} = \frac{\sum fX}{\sum f} $

• Direct Method (Continuous Series): Here, the mid-point (m) of each class interval is used as X.

  $ \bar{X} = \frac{\sum fm}{\sum f} $

• The Assumed Mean and Step Deviation methods can also be applied to grouped data by incorporating the frequency (f) in the formulas:

  Assumed Mean: $ \bar{X} = A + \frac{\sum fd}{\sum f} $

  Step Deviation: $ \bar{X} = A + \frac{\sum fd'}{\sum f} \times c $

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Weighted Arithmetic Mean

Sometimes, it is necessary to assign different levels of importance, or "weights" (W), to different observations. The Weighted Arithmetic Mean is calculated as:

$ \bar{X}_w = \frac{\sum WX}{\sum W} $

For example, when calculating an average price increase, more weight might be given to essential commodities that form a larger part of a consumer's budget.

Median

The Median is the positional value of a variable that divides the distribution into two equal parts. One part comprises all values greater than or equal to the median, and the other comprises all values less than or equal to it. In simple terms, it is the "middle" element when the data is arranged in ascending or descending order.

A key property of the median is that it is not affected by extreme values (outliers), making it a better measure of central tendency for skewed data compared to the mean.

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Computation of Median

1. Ungrouped Data

To find the median, first, arrange the data in ascending or descending order. The position of the median is found using the formula:

Position of Median = Size of $ \left( \frac{N+1}{2} \right)^{th} $ item, where N is the number of items.

• If N is an odd number, the median is the middle value.
  Example: For the data 1, 3, 4, 6, 8, 10, 12, the median is the 4th item, which is 6.
• If N is an even number, there will be two middle values. The median is the arithmetic mean of these two middle values.
  Example: For the data 25, 28, ..., 45, 46, ..., 65, 72, the median is $ \frac{45+46}{2} = 45.5 $.

2. Discrete Series

For a discrete series, first calculate the cumulative frequency (c.f.). Locate the position of the $ \left( \frac{N+1}{2} \right)^{th} $ item in the cumulative frequency column. The corresponding value of the variable is the median.

3. Continuous Series

For a continuous series, first identify the median class, which is the class where the $ \left( \frac{N}{2} \right)^{th} $ item lies. The median is then calculated using the following formula:

$ Median = L + \frac{\frac{N}{2} - c.f.}{f} \times h $

where:

• $L$ = Lower limit of the median class.
• $N$ = Total frequency ($\sum f$).
• $c.f.$ = Cumulative frequency of the class preceding the median class.
• $f$ = Frequency of the median class.
• $h$ = Magnitude (width) of the median class interval.

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Quartiles and Percentiles

• Quartiles are measures that divide the data into four equal parts.
  • First Quartile ($Q_1$): 25% of the items are below it. $Q_1 = \text{size of } \left( \frac{N+1}{4} \right)^{th} \text{ item}$.
  • Second Quartile ($Q_2$): This is the Median itself.
  • Third Quartile ($Q_3$): 75% of the items are below it. $Q_3 = \text{size of } 3 \left( \frac{N+1}{4} \right)^{th} \text{ item}$.
• Percentiles divide the distribution into one hundred equal parts (e.g., $P_1, P_2, \dots, P_{99}$). $P_{50}$ is the median.

Mode

The Mode is the value that occurs most frequently in a data series. It represents the most typical or fashionable value in a distribution. It is denoted by $M_o$. The mode is particularly useful for describing qualitative data or for identifying the most popular item in a set.

Example: For a shoe manufacturer, the modal shoe size is the most important average because it indicates the size with the maximum demand.

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Computation of Mode

1. Discrete Series

In a discrete series, the mode is simply the value with the highest frequency. A distribution can be:

• Unimodal: Has one mode. (e.g., 1, 2, 3, 4, 4, 5)
• Bimodal: Has two modes. (e.g., 1, 2, 2, 3, 4, 4, 5)
• Multimodal: Has more than two modes.
• No Mode: If no value appears more frequently than any other.

2. Continuous Series

For a continuous series, first identify the modal class, which is the class with the highest frequency. The mode is then calculated using the following formula:

$ M_o = L + \frac{D_1}{D_1 + D_2} \times h $

where:

• $L$ = Lower limit of the modal class.
• $D_1$ = Difference between the frequency of the modal class and the frequency of the preceding class.
• $D_2$ = Difference between the frequency of the modal class and the frequency of the succeeding class.
• $h$ = Class interval of the modal class.

To use this formula, the class intervals should be equal and the series should be in an exclusive form.

Relative Position of Mean, Median, and Mode

The relative position of the Arithmetic Mean ($\bar{X}$), Median ($M_i$), and Mode ($M_o$) depends on the shape of the frequency distribution.

• Symmetrical Distribution: In a perfectly symmetrical distribution (like a bell-shaped curve), the Mean, Median, and Mode are all equal.

  $ \bar{X} = M_i = M_o $

• Asymmetrical (Skewed) Distribution: In a skewed distribution, these three values are not equal. The median is always located between the arithmetic mean and the mode.
  • For a positively skewed distribution (tail to the right): $ \bar{X} > M_i > M_o $
  • For a negatively skewed distribution (tail to the left): $ \bar{X} < M_i < M_o $

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Conclusion: Choosing the Right Average

Measures of central tendency summarise a dataset into a single, most representative value. The choice of which average to use is crucial and depends on the purpose of the analysis and the nature of the data distribution.

• The Arithmetic Mean is simple to calculate and is based on all observations, but it is heavily influenced by extreme values (outliers).
• The Median is a better summary for data with extreme values as it is a positional average and not affected by them. It is also suitable for open-ended distributions.
• The Mode is the most appropriate measure for describing qualitative data or identifying the most common value.

Both the Median and Mode can be determined graphically, which is another advantage in certain contexts.

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