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Class 11 Commerce

Economics – Statistics for Economics

Measures of Central Tendency

Measures of Central Tendency

Action PlanNotesQuestionsMock-Test
Study Action Plan (Sorted by Difficulty)
Continuous and Discrete Series
Accuracy: 0%
Total Attempted0
Total Correct0
Total Incorrect0
Difficulty Level:
High
Median
Accuracy: 0%
Total Attempted0
Total Correct0
Total Incorrect0
Difficulty Level:
High
Mode
Accuracy: 0%
Total Attempted0
Total Correct0
Total Incorrect0
Difficulty Level:
High
Quartiles
Accuracy: 0%
Total Attempted0
Total Correct0
Total Incorrect0
Difficulty Level:
High
Weighted Arithmetic Mean
Accuracy: 0%
Total Attempted1
Total Correct0
Total Incorrect1
Difficulty Level:
High
Arithmetic Mean
Accuracy: 75%
Total Attempted4
Total Correct3
Total Incorrect1
Difficulty Level:
Low
Calculation Methods for Arithmetic Mean
Accuracy: 100%
Total Attempted2
Total Correct2
Total Incorrect0
Difficulty Level:
Low
Measures of Central Tendency
Accuracy: 100%
Total Attempted1
Total Correct1
Total Incorrect0
Difficulty Level:
Low

Learning Objectives

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Learning Objectives

  • Understand the need for summarising a set of data by one single number.
  • Recognise and distinguish between the different types of averages.
  • Learn to compute different types of averages.
  • Draw meaningful conclusions from a set of data.
  • Develop an understanding of which type of average would be the most useful in a particular situation.

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Revision Notes & Summary

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Measures of Central Tendency

Introduction

  • Measures of central tendency summarize a set of data by a single representative value.
  • Common examples include average marks, average rainfall, and average income.

Types of Averages

  1. Arithmetic Mean
    • Defined as the sum of all observations divided by the number of observations.
    • Example: For marks 40, 50, 55, 78, 58, the mean is calculated as:
      X=40+50+55+78+585=56.2X = \frac{40 + 50 + 55 + 78 + 58}{5} = 56.2
  2. Median
    • The middle value when data is arranged in order.
    • Example: For the data set 1, 3, 4, 5, 6, 7, 8, 10, 12, the median is 6.
  3. Mode
    • The value that occurs most frequently in a data set.
    • Example: In the data set 1, 2, 3, 4, 4, 5, the mode is 4.

Properties of Averages

  • The sum of deviations from the arithmetic mean is always zero.
  • The arithmetic mean is sensitive to extreme values.
  • The median is less affected by outliers and is a better measure for skewed distributions.

Calculation Methods

Arithmetic Mean

  • Ungrouped Data: Direct method or assumed mean method.
  • Grouped Data: Direct method and step deviation method.

Median Calculation

  • For ungrouped data, sort the data and find the middle value.
  • For grouped data, use cumulative frequency to locate the median class.

Mode Calculation

  • Can be computed for both discrete and continuous data.
  • Data can be unimodal, bimodal, or multimodal.

Conclusion

  • Selecting the appropriate average depends on the nature of the data and the purpose of analysis.
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Practice Test – MCQs, True/False

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Multiple Choice Questions

A.

Arithmetic Mean

B.

Median

C.

Mode

D.

Geometric Mean
Correct Answer: C

Solution:

Mode is the most appropriate measure for determining the most common shoe size, as it represents the value that appears most frequently.

Chapter Concept:

Measures of Central Tendency

A.

It is always equal to the median.

B.

It is affected by extreme values.

C.

It is always less than the mode.

D.

It is not affected by the number of observations.
Correct Answer: B

Solution:

The arithmetic mean is affected by extreme values because it considers all data points in its calculation.

Chapter Concept:

Continuous and Discrete Series

A.

ΣWXΣW\frac{\Sigma WX}{\Sigma W}

B.

ΣXN\frac{\Sigma X}{N}

C.

ΣfXΣf\frac{\Sigma fX}{\Sigma f}

D.

ΣdN\frac{\Sigma d}{N}
Correct Answer: A

Solution:

The weighted arithmetic mean is calculated using the formula ΣWXΣW\frac{\Sigma WX}{\Sigma W}, where WW represents the weights assigned to each data point.

Chapter Concept:

Weighted Arithmetic Mean

A.

Store A

B.

Store B

C.

Both stores will have the same arithmetic mean

D.

Cannot be determined from the given data
Correct Answer: A

Solution:

The arithmetic mean of Store A's sales is 100+150+200+250+3005=200\frac{100 + 150 + 200 + 250 + 300}{5} = 200 units. Increasing this by 10% gives 200+0.1×200=220200 + 0.1 \times 200 = 220 units. The arithmetic mean of Store B's sales is 120+180+240+300+3605=240\frac{120 + 180 + 240 + 300 + 360}{5} = 240 units. Decreasing this by 10% gives 240−0.1×240=216240 - 0.1 \times 240 = 216 units. Therefore, Store A will have a higher arithmetic mean after the adjustment.

Chapter Concept:

Arithmetic Mean

A.

Direct Method

B.

Assumed Mean Method

C.

Step Deviation Method

D.

None of the above
Correct Answer: A

Solution:

In a continuous series, using mid-points for calculating the arithmetic mean is a part of the Direct Method.

Chapter Concept:

Measures of Central Tendency

A.

The IQR will increase.

B.

The IQR will decrease.

C.

The IQR will remain the same.

D.

The IQR will become zero.
Correct Answer: C

Solution:

The interquartile range (IQR) is calculated as Q3 - Q1. Adding a value outside the range of Q1 and Q3 does not affect the IQR, so it remains the same.

Chapter Concept:

Quartiles

A.

25%

B.

50%

C.

75%

D.

100%
Correct Answer: B

Solution:

The central 50% of the data lies between the first quartile (Q1) and the third quartile (Q3).

Chapter Concept:

Quartiles

A.

5

B.

6

C.

7

D.

8
Correct Answer: B

Solution:

Arranging the data in ascending order gives: 1, 3, 4, 5, 6, 7, 8, 10, 12. The middle value is 6.

Chapter Concept:

Median

A.

33

B.

35

C.

45.5

D.

46
Correct Answer: C

Solution:

With 20 values, the median is the average of the 10th and 11th values: (45 + 46) / 2 = 45.5.

Chapter Concept:

Median

A.

By using mid-points of class intervals

B.

By using the frequency of each observation

C.

By assuming a mean and calculating deviations

D.

By using the sum of all observations
Correct Answer: A

Solution:

In a continuous series, the arithmetic mean is calculated using the mid-points of class intervals, which are then used in the formula for mean.

Chapter Concept:

Calculation Methods for Arithmetic Mean

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Measures of Central Tendency

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True or False

Correct Answer: True

Solution:

The arithmetic mean for a discrete series is calculated using the formula Xˉ=ΣfXΣf\bar{X} = \frac{\Sigma fX}{\Sigma f}, where ΣfX\Sigma fX is the sum of the product of observations and their frequencies, and Σf\Sigma f is the sum of frequencies.

Chapter Concept :

Continuous and Discrete Series

Correct Answer: True

Solution:

The step deviation method simplifies calculations by dividing all deviations from the assumed mean by a common factor, reducing the size of numerical figures.

Chapter Concept :

Calculation Methods for Arithmetic Mean

Correct Answer: True

Solution:

The excerpt explains that the arithmetic mean can be calculated using the direct method, assumed mean method, and step deviation method, each suitable for different data types.

Chapter Concept :

Calculation Methods for Arithmetic Mean

Correct Answer: True

Solution:

In continuous series, the mid-points of class intervals are indeed used to calculate the arithmetic mean, as stated in the text.

Chapter Concept :

Continuous and Discrete Series

Correct Answer: True

Solution:

In the assumed mean method, any value, whether existing in the data or not, can be taken as the assumed mean to simplify calculations.

Chapter Concept :

Calculation Methods for Arithmetic Mean

Correct Answer: True

Solution:

The mode is defined as the value that appears most frequently in a data set, making it a measure of central tendency.

Chapter Concept :

Mode

Correct Answer: False

Solution:

The direct method is suitable for smaller data sets. For larger data sets with large numerical figures, the assumed mean method or step deviation method is more appropriate.

Chapter Concept :

Calculation Methods for Arithmetic Mean

Correct Answer: True

Solution:

The arithmetic mean is affected by extreme values because it is calculated as the sum of all observations divided by the number of observations. Therefore, any large or small value can significantly influence the mean.

Chapter Concept :

Measures of Central Tendency

Correct Answer: True

Solution:

The mode is defined as the value that appears most frequently in a data set. It can indeed be unimodal, bimodal, or multimodal depending on the number of values that appear with the highest frequency.

Chapter Concept :

Mode

Correct Answer: True

Solution:

For continuous series, the arithmetic mean is calculated using the mid-points of various class intervals.

Chapter Concept :

Arithmetic Mean

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