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Chapter Analysis
Intermediate15 pages • EnglishQuick Summary
The chapter on Measures of Central Tendency introduces the concept of summarising data with a single representative value using statistical averages. It covers three primary measures: arithmetic mean, median, and mode, each serving different analytical purposes depending on the data distribution. The chapter explains the calculation methods for each measure, including the direct method, assumed mean method, and step deviation method and their application for both grouped and ungrouped data. It also discusses the importance of choosing the appropriate average depending on data characteristics and analysis objectives.
Key Topics
- •Arithmetic Mean
- •Median
- •Mode
- •Direct Method
- •Assumed Mean Method
- •Step Deviation Method
- •Grouped data
- •Ungrouped data
Learning Objectives
- ✓Understand the need for summarising a set of data by one single number.
- ✓Recognize and distinguish between different types of averages.
- ✓Learn to compute different types of averages.
- ✓Draw meaningful conclusions from a set of data.
- ✓Develop understanding of which type of average is the most useful in a particular situation.
Questions in Chapter
Which average would be suitable in the following cases? (i) Average size of readymade garments. (ii) Average intelligence of students in a class. (iii) Average production in a factory per shift. (iv) Average wage in an industrial concern. (v) When the sum of absolute deviations from average is least. (vi) When quantities of the variable are in ratios. (vii) In case of open-ended frequency distribution.
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Indicate the most appropriate alternative from the multiple choices provided against each question.
Answer: (i) b (ii) c (iii) b
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Comment whether the following statements are true or false. (i) The sum of deviation of items from median is zero. (ii) An average alone is not enough to compare series. (iii) Arithmetic mean is a positional value. (iv) Upper quartile is the lowest value of top 25% of items. (v) Median is unduly affected by extreme observations.
Answer: (i) False (ii) True (iii) False (iv) True (v) False
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If the arithmetic mean of the data given below is 28, find (a) the missing frequency, and (b) the median of the series.
Answer: The value of missing frequency is 20 and value of the median is Rs 27.41
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The following table gives the daily income of ten workers in a factory. Find the arithmetic mean.
Answer: Rs 240
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Following information pertains to the daily income of 150 families. Calculate the arithmetic mean.
Answer: Rs 116.3
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Additional Practice Questions
Define 'Measures of Central Tendency'. Explain its importance in data analysis.
mediumAnswer: Measures of Central Tendency are statistical metrics that summarize a set of data points by identifying the central or typical value in the dataset. They hold significant importance in data analysis as they provide a quick summary of the dataset, facilitating easier comparison between different datasets and aiding in clearing insights about the nature of the data.
How is the arithmetic mean calculated in a discrete series?
easyAnswer: In a discrete series, arithmetic mean is calculated by multiplying the frequency of each observation by its value, summing all the products, and dividing by the total number of frequencies.
Explain the difference between the median and the mode.
easyAnswer: The median is the middle value that separates the higher half from the lower half of the data set, while the mode is the value that appears most frequently in the dataset. Thus, median focuses on the positional value, while the mode outlines the most popular or frequent value.
What is the step deviation method and when is it used?
mediumAnswer: The step deviation method is a simplified mean calculation technique that converts large numerical frequency data into manageable figures by dividing deviations of observations from the assumed mean by a common factor. It's primarily used when data involves repetitive calculations with large values.
In what scenario could using the mode be favored over the arithmetic mean or the median?
mediumAnswer: The mode is preferred when the most frequent or typical value of a dataset is needed, such as in determining the most common shoe size or most common examination score in a repetitive dataset. It is particularly effective for qualitative data where category counts matter.
Illustrate a case where the arithmetic mean might not be the best measure of central tendency.
hardAnswer: The arithmetic mean may not be ideal in skewed distributions or when the dataset contains outliers. For example, in income data with extremely high salaries, the mean can be misleadingly high, giving a sense of higher average income than the reality for most of the population. Here, the median would give a better central tendency.
Describe how weights affect the weighted arithmetic mean.
hardAnswer: In a weighted arithmetic mean, different data points are multiplied by predetermined weights which reflect their importance in the dataset. The sum of weighted data points is divided by the sum of weights, prioritizing more significant observations and giving a better reflection of their impact.
What conceptual misunderstanding could arise from misinterpreting the mean of a dataset?
hardAnswer: A common misconception is believing the mean to represent a typical value for all observations, whereas it can be significantly affected by extreme outliers, leading to a misleading representation of 'central' data in skewed distributions, distorting real-world applicability.
Why is it important to distinguish between different types of averages?
mediumAnswer: Different types of averages inform different aspects of the data: the mean provides an overall average, the median offers a middle-ground unaffected by extremes, and the mode highlights the most common values. Choosing the right average depends on the context and the specific details one seeks to analyze.
When and why would you use the median instead of the mean?
mediumAnswer: The median is used instead of the mean in skewed distributions or when the dataset contains outliers since the median is not affected by extreme values. It provides a more accurate representation of central tendency in such cases.