Measures of Central Tendency Class 11 Notes: CBSE (Statistics for Economics) Chapter 5

A. Arithmetic Mean (AM)

The arithmetic mean is the central value obtained by summing all data points and dividing the total by the number of items.

• Simple Arithmetic Mean: All items are assigned equal importance.

• Weighted Arithmetic Mean: Different items are weighted based on their relative importance.

Arithmetic Mean (AM):

$\bar{x} = \frac{\sum x_i}{N}$

$\text{Mean} = \frac{\sum f_i x_i}{\sum f_i}$

Formulae of Calculating Arithmetic Mean:

Combines Mean: $\bar{X}_{12} = \frac{\bar{X}_1 N_1 + \bar{X}_2 N_2}{N_1 + N_2}$

Weighted Mean:  $\bar{X} = \frac{\sum WX}{\sum W}$

Merits and Demerits of Arithmetic Mean:

Merits:

• Simple to calculate.

• Based on all values.

• Suitable for algebraic treatment.

• Accurate and reliable for comparison.

Demerits:

• Affected by extreme values.

• May not represent qualitative data.

• This can lead to misleading conclusions.

A. Median:

The median is the middle value of a series when arranged in ascending or descending order.

$\text{Median} = L + \left( \frac{\frac{N}{2} - F}{f} \right) \cdot h$

Partition Values:

• Partition values divide the data into specific segments:

• Quartiles: Divide data into four equal parts.

• Deciles: Divide data into ten equal parts.

• Percentiles: Divide data into 100 equal parts.

Quartiles:

• First Quartile (Q1): Median of the lower half of the data; 25% of values lie below Q1.

• Second Quartile (Q2): Also called the Median; 50% of values lie below and above Q2.

• Third Quartile (Q3): Median of the upper half of the data; 75% of values lie below Q3.

Where:

• L: Lower boundary of the median class

• N: Total frequency

• F: Cumulative frequency before the median class

• f: Frequency of the median class

• h: Class width

Merits and Demerits of Median

Merits:

• Simple to compute and unaffected by extreme values.

• Useful when data is incomplete.

• Provides a clear positional measure.

Demerits:

• Not based on all data points.

• Requires data arrangement, which can be time-consuming.

• Limited algebraic treatment.

A. Mode:

The mode is the value that occurs most frequently in a dataset and has the highest frequency.

Calculating Mode in Continuous Series:

Mode = $L_1 + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times C$ 

Where, $L_1 = $ Lower limit of the modal class 

$f_0 = $ Frequency of the group preceding the modal class 

Merits and Demerits of Mode:

Merits:

• Simple and widely used.

• Less influenced by marginal values.

• Can be represented graphically.

Demerits:

• Uncertain and challenging for multi-modal datasets.

• Not based on all observations.

• Limited algebraic application.

Relation Between Mean, Median, and Mode:

Mode = $3 \cdot \text{Median} - 2 \cdot \text{Mean}$ 

Mode = $L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \cdot h$

Where:

• L: Lower boundary of the modal class

• f1​: Frequency of the modal class

• f0​: Frequency of the class preceding the modal class

• f2​: Frequency of the class succeeding the modal class

• h: Class width

Graphical Methods for Median and Mode

Locating the Median:

1. Using Cumulative Frequency (Ogives):

• Convert data to a "less than" or "more than" cumulative frequency series.

• Identify N/2N/2N/2 on the y-axis and draw a perpendicular to the curve.

• The intersection with the x-axis gives the median.

2. Using Less Than and More Than Ogives:

• Draw both ogives simultaneously.

• The point where the curves intersect corresponds to the median.

Graphical Presentation of Mode:

1. Construct a histogram of the data.

2. Identify the modal class (the tallest rectangle).

3. Draw diagonals from:

• The top-left of the modal class to the top-left of the following class.

• The top-right of the modal class to the top-right of the preceding class.

4. The intersection of these lines determines the mode on the x-axis.

What is the Purpose and Functions of Averages ?

1. Represents data concisely.

2. Enables comparisons across datasets.

3. Aids in policy formulation.

4. Forms the basis for statistical analysis.

5. Provides a single representative value for a group.

Importance of A  Good Average:

1. Easy to compute and understand.

2. Clearly defined and universally accepted.

3. Based on all items in the series.

4. Consistent and stable.

5. Least affected by sample changes.

6. Capable of algebraic manipulation.

Benefits of Vedantu’s CBSE Class 11 Measures of Central Tendency  Notes

• Simplified explanations of mean, median, and mode ensure a solid grasp of key topics.

• Worked examples guide students in solving problems efficiently.

• Real-life applications of central tendencies enhance comprehension.

• Highlights important questions and frequently tested concepts.

• Downloadable resources ensure revision convenience anytime, anywhere.

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Conclusion

Measures of Central Tendency, discussed in Chapter 5 of Class 11 Statistics Notes, form the cornerstone of statistical analysis. By mastering mean, median, and mode, students can effectively analyse datasets and draw meaningful conclusions. These concepts not only enhance statistical understanding but also find practical applications in diverse fields, making this chapter a vital part of the CBSE curriculum.   With expert guidance, time-saving features, and the convenience of online access, Vedantu’s notes ensure that students are well-equipped for exams and beyond.

Students can also visit and refer to other study materials of Economics Indian Economic Development Notes for better exam preparations and to achieve good scores as this content is created by Vedantu experts.

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