Biostatistics: Mean for Frequency Tables
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Biostatistics: Mean for Frequency Tables

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Questions and Answers

What does a measure of dispersion indicate about a set of data?

  • The central tendency of the values
  • The frequency of the values
  • The predictability of the values
  • The variability present in the values (correct)

    • Which of the following is NOT a measure of dispersion?

  • Coefficient of variation
  • Variance
  • Standard deviation
  • Median (correct)

    • When is there no dispersion in a data set?

  • When all the values are the same (correct)
  • When the values are closely spaced
  • When the values are widely scattered
  • When all the values are different

    • How is the range (R) of a data set calculated?

    <p>Largest value - Smallest value</p> Signup and view all the answers

    What does a small amount of dispersion indicate about values in a data set?

    <p>Values are close to each other</p> Signup and view all the answers

    What is the formula for calculating sample variance?

    <p>$S^2 = \frac{\sum_{i=1}^{n} (x_i - x̄)^2}{n-1}$</p> Signup and view all the answers

    How is the population variance calculated?

    <p>$\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - μ)^2}{N}$</p> Signup and view all the answers

    What is true about the relationship between standard deviation and variance?

    <p>Standard deviation is the square root of variance.</p> Signup and view all the answers

    In which situation would the coefficient of variation be most useful?

    <p>Comparing two datasets with different units of measurement.</p> Signup and view all the answers

    Which of the following correctly represents the coefficient of variation (C.V)?

    <p>$C.V = \frac{S}{X̄}(100)$</p> Signup and view all the answers

    Study Notes

    Biostatistics Lectures

    • Lectures were prepared by Assoc. Prof. M A Alawady, Department of Mathematics, Faculty of Science, Zagazig University in 2022.

    Mean for Organized Data in a Frequency Table

    • Data for a discrete variable is given in a frequency table (i, values, frequency).

    • Mean (denoted by ) is calculated using the formula:

      = Σ(fᵢ x xᵢ) / n where fᵢ = frequency of a value, xᵢ = value, and n= total frequency (Σfᵢ) .

    Example of Calculating Mean

    • Data: Number of subjects where students failed.
    • Frequency table:
      • i | No. of subjects failed | Frequency
      • 1 | 0 | 8
      • 2 | 1 | 18
      • 3 | 2 | 12
      • 4 | 3 | 2
    • Total = 40
    • Mean calculation : = (08 + 118 + 212 + 32) / 40 = 48/40 = 1.2

    Mean for Organized Data in a Frequency Distribution Table

    • Data for a continuous variable is given in a frequency distribution table (i, class boundaries, midpoint, frequency, cumulative frequency)

    • Mean (denoted by ) is calculated using the formula:

      = Σ(fᵢ x xᵢ) / n

    Example of Calculating Mean for Continuous Data

    • Data: Mileage of cars per liter of fuel (continuous data).

    • Frequency table:

      • i | Class Boundaries | Class Midpoint | Frequency
      • 1 | 11.5-13.5 | 12.5 | 8
      • 2 | 13.5-15.5 | 14.5 | 10
      • 3 | 15.5-17.5 | 16.5 | 12
      • 4 | 17.5-19.5 | 18.5 | 6
      • 5 | 19.5-21.5 | 20.5 | 4

    • Total = 40

    • Mean calculation : = (12.58 +14.510+16.512+18.56+20.5*4) / 40 = 636/40 = 15.9

    Weighted Mean

    • Weighted mean is used when some observations are more significant than others.
    • Formula: = Σ(wᵢ x xᵢ) / Σwᵢ

    Example of Weighted Mean

    • Student grades in different courses:
    • Formula : GPA = Σ(wᵢ x xᵢ) / Σwᵢ
    • Student grades : Maths (B, 16 points, 4 credit hours), Statistics (A, 15 points, 3 credit hours), English (C, 9 points, 3 credit hours), Physics (C, 12 points, 4 credit hours)
    • Calculation: GPA = 52/14 = 3.71

    Median for Grouped Data

    • For discrete quantitative data with a frequency table
    • Steps:
    1. Calculate the cumulative frequency.
    2. If n is odd, the median is the value with the smallest cumulative frequency greater than or equal to (n+1)/2
    3. If n is even, find the two central values and take their average.

    Example of Calculation of Median

    • Subjects | No. of subjects failed | Frequency
    • Maths | 0 | 8
    • Stats| 1 | 18
    • Eng| 2 | 12
    • Chem| 3 | 2
    • n = 40
    • Median class = 1(cumulative frequency 26)

    Median for Continuous Data

    • Formula: = L + [(n/2) - F] / f x C Where:
      • L = lower limit of the median class
      • F = cumulative frequency of the class before the median class
      • f = frequency of the median class
      • c = class width

    Example of Calculating Median for Continuous Data

    • Calculate the median for the following data:
    • Class boundaries| Midpoint | Frequency | Cumulative Frequency
    • 11.5 - 13.5 | 12.5 | 8 | 8
    • 13.5 - 15.5 | 14.5 | 10 | 18
    • 15.5 - 17.5 | 16.5 | 12 | 30
    • 17.5 - 19.5 | 18.5 | 6 | 36
    • 19.5 - 21.5 | 20.5 | 4 | 40
    • Median class is 15.5-17.5

    Mode for Grouped Data

    • Mode is the value that appears most frequently in the data.
    • For discrete data, find the observation with the highest frequency.
    • For continuous data, find the class with the highest frequency.

    Example of Finding Mode

    • Marks (out of 10) | Frequency
    • 2| 1
    • 3| 2
    • 4| 3
    • 5| 2
    • 6| 3
    • 7| 3
    • 8| 2
    • 9| 4
    • 10 | 2
    • Mode = 9

    Measures of Dispersion

    • Range = Largest value - Smallest value
    • Variance: Measures how spread out the values are from the mean.
    • Sample Variance (S2) = Σ(xᵢ - x̅)2 / (n-1)
    • Population Variance (σ2) = Σ(xᵢ - μ)2 / N
    • Standard Deviation (S or σ): Square root of the variance.
    • Coefficient of Variation (C.V.): (S / x̅) * 100.

    Examples of Measures of Dispersion Calculations

    • Data on subjects failed by students to illustrate the concepts of Range, Variance, Standard Deviation, and Coefficient of Variation calculations

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    Related Documents

    Biostatistics Lectures PDF

    Description

    This quiz covers the calculation of mean for organized data presented in frequency tables, focusing on both discrete and continuous variables. It includes examples and formulas applicable to real data. Test your understanding of biostatistics concepts related to mean calculations.

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