Statistics: Mean, Median, and Mode

Introduction

• Provides a quick revision of mean, median, and mode.
• Focuses on consolidating formulas, identifying important question types, and understanding formula application.
• Aims for a chapter-ready state with clarity on formula selection and use.
• PDF of notes will be provided.

Types of Series

• Understanding individual, discrete, and continuous series is crucial for any statistics chapter.
• Individual Series:
  • Only x values are provided
  • No frequency (f)
• Discrete Series:
  • Both x and f values are provided
• Continuous Series:
  • Both x and f values are provided.
  • X values are in interval form (e.g., 0-10, 10-20)
• Frequency:
  • Represents how many times a particular value repeats.
  • Example: If 4000 is the salary for 25 people, the frequency of 4000 is 25.

Arithmetic Mean

• Three types: simple, combined, and weighted.
• Simple Arithmetic Mean:
  • Has seven formulas.
  • Divided into individual, discrete, and continuous series calculations.
• Combined and Weighted Arithmetic Mean:
  • Each has one formula.
  • Often used in larger, more complex questions.

Simple Arithmetic Mean Formulas

• Individual Series
  • Direct Method: x̄ = Σx / n
    • Σx means the sum of all x values.
    • n is the number of values.
  • Short-cut Method: x̄ = a + Σd / n
    • a is the assumed mean.
    • d is the deviation (x - a).
• Discrete Series
  • Direct Method: x̄ = Σfx / n
  • Short-cut Method: x̄ = a + Σfd / n
• Continuous Series
  • Direct Method: x̄ = Σfm / n
    • m represents the mid-value of the interval.
  • Short-cut Method: x̄ = a + Σfd / n
  • Step Deviation Method: x̄ = a + (Σfd’ / n) * i
    • d' is the step deviation.
    • i is the interval length

Combined Arithmetic Mean

• Used when combining multiple means.
• Two Means: x̄₁₂ = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)
• Three Means: x̄₁₂₃ = (n₁x̄₁ + n₂x̄₂ + n₃x̄₃) / (n₁ + n₂ + n₃)

Weighted Arithmetic Mean

• Used when different data points have different weights.
• Formula: x̄ = Σwx / Σw
  • w represents the weight.

Types of Questions

• Formula-based direct questions
• Less than and More than types of questions
• Unequal class interval questions
• Finding missing values or frequencies
• Correcting incorrect values of the mean

Question Examples

• Example 1: Individual Series, Direct Method
  • Given pocket allowances of 10 students.
  • Calculate arithmetic mean using x̄ = Σx / n
• Example 2: Individual Series, Shortcut Method
  • Same data, but calculate using an assumed mean (a).
  • Use x̄ = a + Σd / n, where d = x - a
• Example 3: Discrete Series, Direct Method
  • Given wages (x) and number of workers (f).
  • Use x̄ = Σfx / n
• Example 4: Discrete Series, Shortcut Method
  • Same data, calculate using shortcut method
  • Use x̄ = a + Σfd / n, where d = x - a
• Example 5: Continuous Series, Direct Method
  • Given class intervals and number of students
  • Use x̄ = Σfm / n, where m is the midpoint of the interval.
• Example 6: Continuous Series, Shortcut Method
  • Same data, calculate using shortcut method : x̄ = a + Σfd / n
• Example 7: Continuous Series, Step Deviation Method
  • Use x̄ = a + (Σfd’ / n) * i, where d' = (x-a)/i
• Handling "Less Than" Data:
  • Create class intervals from the given "less than" values
  • Calculate frequencies by subtracting successive values
• Handling "More Than" Data:
  • Create class intervals using more than values
  • Calculate frequencies by subtracting successive values from bottom

Finding Missing Values in Frequency

• Given data with a missing frequency and the overall mean.
• Set up the equation using the mean formula and solve for the missing frequency.

Correcting Incorrect Values

• Subtract the incorrect values and add the correct values to the sum, then calculate the corrected mean.

Weighted Mean Example

• Given items and their weights.
• Calculate the weighted mean using x̄ = Σwx / Σw

Median Formulas

• Individual Series:
  • m = size of (n + 1)/2 th item
• Discrete Series:
  • m = size of (n + 1)/2 th item
• Continuous Series:
  • m = l1 + (n/2 - cf)/f * i
    • l1 = lower limit of the median class
    • n = total frequency
    • cf = cumulative frequency of the class preceding the median class
    • f = frequency of the median class
    • i = class interval size

Key Considerations for Median

• Data needs to be arranged in ascending or descending order before applying formula.
• When frequencies are given, the cumulative frequency (CF) needs to be calculated

Examples

• Example 1: Individual Series
  • Arrange data in ascending order and apply m = size of (n + 1)/2 th item.
  • If (n+1)/2 results in decimal 4.5, then average fourth and fifth items
• Example 2: Discrete Series
  • Calculate cumulative frequency. m = size of (n + 1)/2 th item
  • Identify the value in CF which is just greater than (n+1)/2.
  • Corresponding X value is the median
• Example 3: Continuous Series
  • Calculate Cumulative frequency, find N/2
  • Identify the class in cf just greather then N/2, apply m = l1 + (n/2 - cf)/f * i

Mode

• The value that appears most frequently in a data set
• Individual -- Find through simple inspection
• Discrete Series -- Typically solved through inspection
• Continuous Series -- use the formula z = l1 + (f1 - f0)/ 2f1 - f0 - f2*i

Examples (Focusing on Continuous Series)

• Example: If, after inspecting data, the mode is 30 (value 15-20) and the corresponding items: _l1 (Lower Class limit)= 15 (Lower item in mode. from "value" -- (15-20) _f1 = 30 (frequency in z) _f0 = 15-20: (frequency PRECEDING f1 (frequency) _f0 = 20-25: (frequency FOLLOWING f1 (frequency)