Statistics- Mean, Mode & Median


  Mean, Mode, and Median
  
    In this chapter, we explore the calculations of the mean, mode, and median.
  
Mean
 For the mean, we discuss three methods:
  
    Direct Method
    Step Deviation Method
    Assumed Mean Method
  
  
    We also explain the various symbols used:
  
  
    x represents the observed value (or class representative for grouped data).
    f denotes the frequency of each class interval.
    d represents the deviation of x (or class representative) from a reference value. In the step deviation method, it is computed as d = (x − a) / h.
    a is the assumed mean, selected close to the expected mean.
    h is the step size (or class width) used to simplify calculations.
  

  
  Direct Method
  
    The Direct Method calculates the mean (μ) by summing all values (or the products of frequency and representative value for grouped data) and dividing by the total frequency.
  
  Formula for ungrouped data:
   $μ = \frac{\sum x}{n}$ 
  For grouped data:
   $μ = (\sum f$ _ix_i)/(∑f_i)
  Example: Direct Method Using a Frequency Distribution Table
  Consider the following frequency distribution table:
  
    
      
        Class Interval
        Frequency (f)
        Representative Value (x)
        f × x
      
    
    
      
        10 – 20
        4
        15
        4 × 15 = 60
      
      
        20 – 30
        6
        25
        6 × 25 = 150
      
      
        30 – 40
        5
        35
        5 × 35 = 175
      
      
        40 – 50
        3
        45
        3 × 45 = 135
      
      
        Total
        18
        
        520
      
    
  
  
    Thus, the mean is calculated as:
  
   $μ = \frac{520}{18}$ 
  
    Mean ≈ 28.89
  

  
  Step Deviation Method
  
    This method is particularly useful when the representative values are large. An assumed mean a (usually one of the representative values) and a step size h (class width) are selected. The deviation d is computed by:
  
  
    d = (x − a) / h
  
  
    Then, the mean is computed using:
  
   $μ = a + h \cdot \frac{\sum f d}{n}$ 
  Example: Step Deviation Method Using a Frequency Distribution Table
  Consider the following table. Assume a = 35 (chosen representative value) and h = 10:
  
    
      
        Class Interval
        Frequency (f)
        Representative Value (x)
        x − a
        d = (x − a) / h
        f × d
      
    
    
      
        20 – 30
        5
        25
        25 − 35 = -10
        -10 / 10 = -1
        5 × (-1) = -5
      
      
        30 – 40
        12
        35
        35 − 35 = 0
        0 / 10 = 0
        12 × 0 = 0
      
      
        40 – 50
        8
        45
        45 − 35 = 10
        10 / 10 = 1
        8 × 1 = 8
      
      
        Total
        25
        
        
        
        3
      
    
  
  
    The mean is then:
  
   $μ = a + h \cdot \frac{3}{25} = 35 + \frac{30}{25}$ 
  
    Mean ≈ 35 + 1.2 = 36.2
  

  
  Assumed Mean Method
  
    The Assumed Mean Method is similar to the step deviation method but uses the simple deviation d = x − a (without dividing by h). It is particularly useful for mental calculations.
  
  
    The mean is computed as:
  
   $μ = a + \frac{\sum f (x - a)}{n}$ 
  Example: Assumed Mean Method Using a Frequency Distribution Table
  Consider the following table. Assume a = 35:
  
    
      
        Class Interval
        Frequency (f)
        Representative Value (x)
        x − a
        f × (x − a)
      
    
    
      
        20 – 30
        5
        25
        25 − 35 = -10
        5 × (-10) = -50
      
      
        30 – 40
        12
        35
        35 − 35 = 0
        12 × 0 = 0
      
      
        40 – 50
        8
        45
        45 − 35 = 10
        8 × 10 = 80
      
      
        Total
        25
        
        
        30
      
    
  
  
    The mean is then:
  
   $μ = a + \frac{30}{25} = 35 + 1.2$ 
  
    Mean ≈ 36.2
  

  
  Mode
  
    The mode is the value (or class interval) that occurs most frequently in a data set.
  
  
    For grouped data, the modal value is calculated using the Modal Formula:
  
  
    Formula:
  
   $Mode = L + \frac{f₁ - f₀}{2 f₁ - f₀ - f₂} c$ 
  
    Where:
  
  
    L is the lower limit of the modal class.
    f₁ is the frequency of the modal class.
    f₀ is the frequency of the class preceding the modal class.
    f₂ is the frequency of the class succeeding the modal class.
    c is the class width.
  
  Example: Mode Using a Frequency Distribution Table
  Consider the following table:
  
    
      
        Class Interval
        Frequency (f)
      
    
    
      
        10 – 20
        4
      
      
        20 – 30
        6
      
      
        30 – 40
        9
      
      
        40 – 50
        5
      
    
  
  
    The modal class is 30 – 40 since it has the highest frequency (f₁ = 9). Assuming the previous frequency (f₀) is 6, the next frequency (f₂) is 5, the lower limit L = 30, and the class width c = 10, we have:
  
   $Mode = 30 + \frac{9 - 6}{2 \cdot 9 - 6 - 5} \cdot 10$ 
  
    Simplifying:
  
   $Mode = 30 + \frac{3}{18} \cdot 10 = 30 + 1.67$ 
  
    Mode ≈ 31.67
  

  
  Median
  
    The median is the middle value of an ordered data set. For grouped data, the median is estimated using the cumulative frequency distribution.
  
  For ungrouped data:
  
    
      If the number of observations (n) is odd, the median is the ((n + 1) / 2)^th value.
    
    
      If n is even, the median is the average of the (n/2)^th and ((n/2) + 1)^th values.
    
  
  For grouped data, the median is given by:
   $Median = L + \frac{(n/2) − F}{f} \cdot c$ 
  
    Where:
  
  
    L is the lower limit of the median class.
    n is the total frequency.
    F is the cumulative frequency of the class preceding the median class.
    f is the frequency of the median class.
    c is the class width.
  
  Example: Median Using a Frequency Distribution Table
  Consider the following frequency distribution table:
  
    
      
        Class Interval
        Frequency (f)
        Cumulative Frequency
      
    
    
      
        10 – 20
        4
        4
      
      
        20 – 30
        6
        4 + 6 = 10
      
      
        30 – 40
        9
        10 + 9 = 19
      
      
        40 – 50
        5
        19 + 5 = 24
      
    
  
  
    Here, n = 24, so n/2 = 12. The cumulative frequency first reaches or exceeds 12 in the 30 – 40 class. For this median class:
  
  
    L = 30
    The cumulative frequency of the previous class F = 10
    f = 9
    c = 10 (class width)
  
  
    Substituting these values in the median formula:
  
   $Median = 30 + \frac{12 - 10}{9} \cdot 10$ 
  
    Simplifying:
  
   $Median = 30 + \frac{2}{9} \cdot 10 = 30 + \frac{20}{9}$ 
  
    Median ≈ 32.22
  
  
  
    This chapter demonstrated how to compute the mean using three different methods along with the formulas and tables for mode and median (for both ungrouped and grouped data).
Class Interval	Frequency (f)	Representative Value (x)	f × x
10 – 20	4	15	4 × 15 = 60
20 – 30	6	25	6 × 25 = 150
30 – 40	5	35	5 × 35 = 175
40 – 50	3	45	3 × 45 = 135
Total	18		520
Class Interval	Frequency (f)	Representative Value (x)	x − a	d = (x − a) / h	f × d
20 – 30	5	25	25 − 35 = -10	-10 / 10 = -1	5 × (-1) = -5
30 – 40	12	35	35 − 35 = 0	0 / 10 = 0	12 × 0 = 0
40 – 50	8	45	45 − 35 = 10	10 / 10 = 1	8 × 1 = 8
Total	25				3
Class Interval	Frequency (f)	Representative Value (x)	x − a	f × (x − a)
20 – 30	5	25	25 − 35 = -10	5 × (-10) = -50
30 – 40	12	35	35 − 35 = 0	12 × 0 = 0
40 – 50	8	45	45 − 35 = 10	8 × 10 = 80
Total	25			30
Class Interval	Frequency (f)	Cumulative Frequency
10 – 20	4	4
20 – 30	6	4 + 6 = 10
30 – 40	9	10 + 9 = 19
40 – 50	5	19 + 5 = 24