class 11 notes

Class 11 Statistics Notes Chapter 5 (Measures of central tendency) – Statistics For Economics Book

Philoid

Philoid

— 7 min read

Statistics For Economics
Detailed Notes with MCQs of a crucial chapter for your upcoming exams: Chapter 5, 'Measures of Central Tendency' from your NCERT Statistics for Economics book. This chapter deals with summarizing data by finding a single value that represents the 'center' or 'typical value' of a dataset. Understanding these measures is fundamental for interpreting statistical data.

Chapter 5: Measures of Central Tendency - Detailed Notes

1. Introduction: What is Central Tendency?

  • Definition: A measure of central tendency (or statistical average) is a single value that attempts to describe a set of data by identifying the central position within that set of data. It provides a concise summary of the overall data distribution.
  • Purpose:
    • To get a single representative value for the entire dataset.
    • To facilitate comparison between different datasets.
    • To aid in decision-making and further statistical analysis.
  • Types: The main types of averages we study are:
    • Mathematical Averages: Arithmetic Mean (A.M.)
    • Positional Averages: Median (M) and Mode (Mo)

2. Arithmetic Mean (A.M.) or Mean (X̄)

  • Definition: The sum of all observations divided by the number of observations. It's the most common type of average.
  • Calculation:
    • a) Individual Series (Ungrouped Data): Data given as individual values (e.g., 10, 15, 12, 18, 20).
      • Direct Method: X̄ = ΣX / N
        • Where ΣX = Sum of all observations, N = Number of observations.
      • Assumed Mean (Short-cut) Method: X̄ = A + (Σd / N)
        • Where A = Assumed Mean (any value, often one from the data), d = X - A (deviations from assumed mean).
    • b) Discrete Series (Ungrouped Frequency Distribution): Data given with frequencies (e.g., Value (X): 10, 15, 20; Frequency (f): 2, 3, 1).
      • Direct Method: X̄ = ΣfX / Σf (or ΣfX / N, where N = Σf)
        • Where ΣfX = Sum of the product of values and their corresponding frequencies, N = Total frequency.
      • Assumed Mean Method: X̄ = A + (Σfd / N)
        • Where A = Assumed Mean (usually one of the X values), d = X - A, Σfd = Sum of the product of frequencies and deviations, N = Total frequency.
      • Step-Deviation Method: X̄ = A + (Σfd' / N) * c
        • Where A = Assumed Mean, d' = d / c = (X - A) / c, c = Common factor among deviations (often the class interval if applicable later), N = Total frequency.
    • c) Continuous Series (Grouped Frequency Distribution): Data given in class intervals (e.g., 0-10, 10-20 with frequencies).
      • First Step: Find the mid-point (m) or class mark for each class interval. m = (Lower Limit + Upper Limit) / 2. Then treat 'm' as 'X' from the discrete series.
      • Direct Method: X̄ = Σfm / N (where N = Σf)
      • Assumed Mean Method: X̄ = A + (Σfd / N) (where A = Assumed Mean, usually a mid-point; d = m - A)
      • Step-Deviation Method: X̄ = A + (Σfd' / N) * h (where A = Assumed Mean; d' = (m - A) / h; h = Class Interval width (assuming equal width); N = Total frequency)
  • Properties/Merits:
    • Easy to understand and calculate.
    • Based on all observations in the dataset.
    • Rigidly defined (has a precise mathematical formula).
    • Capable of further algebraic treatment (e.g., calculating combined mean).
    • Relatively stable measure (less affected by sampling fluctuations).
  • Demerits/Limitations:
    • Highly affected by extreme values (outliers).
    • Cannot be calculated for qualitative data (e.g., honesty, beauty).
    • Cannot be calculated for open-end classes without making assumptions.
    • Cannot be determined graphically.
    • May result in a value that doesn't exist in the data (e.g., average 2.5 children).
  • Weighted Arithmetic Mean: Used when different items have different importance (weights).
    • X̄w = ΣwX / Σw
    • Where X = Value of the item, w = Weight assigned to the item.
  • Combined Arithmetic Mean: Mean of a combined group from different subgroups.
    • X̄12 = (N1 * X̄1 + N2 * X̄2) / (N1 + N2)
    • Where X̄1, X̄2 are means of subgroups and N1, N2 are the number of observations in subgroups.

3. Median (M)

  • Definition: The middle value of a dataset when arranged in ascending or descending order. It divides the data into two equal halves. It is a positional average.
  • Calculation:
    • a) Individual Series:
      • Step 1: Arrange data in ascending or descending order.
      • Step 2: Find the position of the median using: Size of (N + 1) / 2 th item.
      • Step 3: If N is odd, Median is the value at this position. If N is even, Median is the average of the values at the N/2 th and (N/2 + 1) th positions.
    • b) Discrete Series:
      • Step 1: Arrange data (X values) in ascending order.
      • Step 2: Calculate Cumulative Frequencies (cf).
      • Step 3: Find the position of the median: Size of (N + 1) / 2 th item (where N = Σf).
      • Step 4: Locate the cf just greater than or equal to (N+1)/2. The value (X) corresponding to this cf is the Median.
    • c) Continuous Series:
      • Step 1: Calculate Cumulative Frequencies (cf).
      • Step 2: Determine the Median Class: The class corresponding to the cumulative frequency in which the N/2 th item lies.
      • Step 3: Apply the formula:
        • M = l + [ (N/2 - cf) / f ] * h
        • Where:
          • l = Lower limit of the median class
          • N = Total frequency (Σf)
          • cf = Cumulative frequency of the class preceding the median class
          • f = Frequency of the median class
          • h = Class interval width of the median class
  • Properties/Merits:
    • Not affected by extreme values.
    • Can be calculated for open-end classes.
    • Can be determined graphically (using Ogives).
    • Suitable for qualitative data that can be ranked.
    • Easy to understand (conceptually).
  • Demerits/Limitations:
    • Requires arrangement of data, which can be time-consuming.
    • Not based on all observations (ignores the magnitude of most values).
    • Not capable of further algebraic treatment (like combined median).
    • Can be less stable than the mean in sampling.
  • Related Positional Values:
    • Quartiles: Divide data into 4 equal parts (Q1, Q2=Median, Q3).
      • Q1 (Lower Quartile) = Size of (N+1)/4 th item (Individual/Discrete); Formula similar to Median for Continuous, using N/4.
      • Q3 (Upper Quartile) = Size of 3(N+1)/4 th item (Individual/Discrete); Formula similar to Median for Continuous, using 3N/4.
    • Deciles: Divide data into 10 equal parts (D1 to D9).
    • Percentiles: Divide data into 100 equal parts (P1 to P99).

4. Mode (Mo)

  • Definition: The value that occurs most frequently in a dataset. It is also a positional average. A dataset can be unimodal (one mode), bimodal (two modes), multimodal (more than two modes), or have no mode.
  • Calculation:
    • a) Individual & Discrete Series:
      • Inspection Method: Identify the value (X) with the highest frequency (f). This value is the Mode.
      • Grouping Method: Used when frequencies are irregular, highest frequency is repeated, or highest frequency is at the beginning/end. Involves creating grouping and analysis tables to ascertain the true mode.
    • b) Continuous Series:
      • Step 1: Identify the Modal Class: The class with the highest frequency (using inspection or grouping method).
      • Step 2: Apply the formula:
        • Mo = l + [ (f1 - f0) / (2f1 - f0 - f2) ] * h
        • Where:
          • l = Lower limit 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 interval width of the modal class
  • Properties/Merits:
    • Not affected by extreme values.
    • Easy to understand and often easy to locate (especially in discrete series).
    • Can be determined graphically (using Histogram - highest bar).
    • Represents the most typical or popular value (useful in business, e.g., shoe size).
    • Can be calculated for open-end classes (if modal class is not open-ended).
    • Can be used for qualitative data.
  • Demerits/Limitations:
    • Not rigidly defined (can be ill-defined if no value repeats or if multiple values have the same highest frequency).
    • Not based on all observations.
    • Not capable of further algebraic treatment.
    • Calculation can be complex with the grouping method.

5. Relationship Between Mean, Median, and Mode

  • For a Symmetrical Distribution: Mean = Median = Mode.
  • For a Moderately Skewed Distribution (Asymmetrical):
    • Positively Skewed (Tail to the right): Mean > Median > Mode
    • Negatively Skewed (Tail to the left): Mean < Median < Mode
    • Empirical Relationship (Approximate): Mode ≈ 3 Median - 2 Mean (Very important for exams!)

6. Choosing an Appropriate Average

  • Mean: Use when data is roughly symmetrical, no extreme outliers, and further algebraic calculations are needed.
  • Median: Use when data is skewed, there are extreme outliers, or dealing with open-end classes or ranked data.
  • Mode: Use when you need the most frequent or typical value, dealing with qualitative data, or for descriptive purposes like finding the most popular item.

Multiple Choice Questions (MCQs)

  1. Which measure of central tendency is most affected by extreme values (outliers)?
    (a) Median
    (b) Mode
    (c) Arithmetic Mean
    (d) Geometric Mean (Not typically covered in detail in this chapter, but A.M. is the primary answer)

  2. The value that divides a dataset into two equal halves after arranging it in order is called:
    (a) Mean
    (b) Mode
    (c) Median
    (d) Range

  3. For a moderately negatively skewed distribution, which of the following is correct?
    (a) Mean = Median = Mode
    (b) Mean > Median > Mode
    (c) Mean < Median < Mode
    (d) Mode = 2 Mean - 3 Median

  4. Which average can be calculated for a continuous series with open-end classes without making assumptions?
    (a) Arithmetic Mean
    (b) Median
    (c) Geometric Mean
    (d) Harmonic Mean

  5. The mode of the data set: 5, 8, 9, 10, 8, 9, 8, 7 is:
    (a) 9
    (b) 5
    (c) 7
    (d) 8

  6. In a continuous frequency distribution, the median class is the class in which the:
    (a) (N+1)/2 th item falls
    (b) N/2 th item falls
    (c) N/4 th item falls
    (d) 3N/4 th item falls

  7. The empirical relationship between Mean, Median, and Mode for a moderately skewed distribution is:
    (a) Mean - Mode = 3 (Mean - Median)
    (b) Mode = 3 Median - 2 Mean
    (c) Median = Mode + Mean / 2
    (d) Both (a) and (b) represent the same relationship rearranged

  8. Which measure of central tendency can be determined graphically using a Histogram?
    (a) Mean
    (b) Median
    (c) Mode
    (d) Quartiles

  9. If the mean of 10 observations is 15, the sum of observations (ΣX) is:
    (a) 1.5
    (b) 25
    (c) 150
    (d) 10

  10. The calculation of Arithmetic Mean for a continuous series requires finding the ______ of each class interval.
    (a) Upper Limit
    (b) Lower Limit
    (c) Frequency
    (d) Mid-point (Class Mark)

Answer Key for MCQs:

  1. (c) Arithmetic Mean
  2. (c) Median
  3. (c) Mean < Median < Mode
  4. (b) Median (Mode can also sometimes be calculated if the modal class is not open-ended, but Median is the most reliable answer here).
  5. (d) 8 (It occurs 3 times, more than any other value).
  6. (b) N/2 th item falls
  7. (d) Both (a) and (b) represent the same relationship rearranged (Check: Mean - Mode = 3 Mean - 3 Median => Mode = Mean - 3 Mean + 3 Median => Mode = 3 Median - 2 Mean. So both are equivalent).
  8. (c) Mode
  9. (c) 150 (Since Mean = ΣX / N => 15 = ΣX / 10 => ΣX = 15 * 10 = 150)
  10. (d) Mid-point (Class Mark)

Study these notes thoroughly. Pay close attention to the formulas, the conditions under which each measure is calculated, and their respective merits and demerits. Understanding the relationship between Mean, Median, and Mode is also frequently tested. Good luck with your preparation!

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