Measures of Central Tendency
What is Measures of Central Tendency?
A single value that represents the center point or typical value of a dataset.
Key formula / rule: Arithmetic Mean (Ungrouped Data)
Key points
- Define and explain the concept of central tendency.
- Calculate the Mean, Median, and Mode for both ungrouped and grouped data.
- Understand the properties, advantages, and limitations of each measure.
- Determine the most appropriate measure of central tendency for a given dataset.
Common exam trap
Not arranging data in order before calculating the Median.
Definitions
- Term
Central Tendency
- Meaning
A single value that represents the center point or typical value of a dataset.
- Term
Mean
- Meaning
The arithmetic average of a set of values, calculated by summing all values and dividing by the number of values.
- Term
Median
- Meaning
The middle value in a dataset when the values are arranged in order.
- Term
Mode
- Meaning
The value that appears most frequently in a dataset.
- Term
Outlier
- Meaning
An observation point that is distant from other observations, potentially distorting the mean.
- Term
Skewness
- Meaning
A measure of the asymmetry of the probability distribution of a real-valued random variable about its mean.
- Term
Frequency Distribution
- Meaning
A table that displays the frequency of various outcomes in a sample, often grouped into classes.
- Term
Cumulative Frequency
- Meaning
The sum of the frequencies for a given class and all preceding classes in a frequency distribution.
Learning objectives
Define and explain the concept of central tendency.
Calculate the Mean, Median, and Mode for both ungrouped and grouped data.
Understand the properties, advantages, and limitations of each measure.
Determine the most appropriate measure of central tendency for a given dataset.
Apply the empirical relationship between Mean, Median, and Mode to solve problems.
Formulae
- Name
Arithmetic Mean (Ungrouped Data)
- Note
Where xᵢ are individual observations and N is the total number of observations.
- Expression
x̄ = (Σxᵢ) / N
- Name
Arithmetic Mean (Grouped Data - Direct Method)
- Note
Where fᵢ is the frequency of class i and xᵢ is the mid-point of class i.
- Expression
x̄ = (Σfᵢxᵢ) / (Σfᵢ)
- Name
Median (Ungrouped Data - N is odd)
- Note
Data must be arranged in ascending or descending order.
- Expression
Median = ((N+1)/2)th observation
- Name
Median (Ungrouped Data - N is even)
- Note
Data must be arranged in ascending or descending order.
- Expression
Median = [ (N/2)th observation + ((N/2)+1)th observation ] / 2
- Name
Median (Grouped Data)
- Note
L = lower boundary of median class, N = total frequency, cf = cumulative frequency of class preceding median class, f = frequency of median class, h = class width.
- Expression
Median = L + [((N/2) - cf) / f] * h
- Name
Mode (Ungrouped Data)
- Note
A dataset can have one, multiple, or no mode.
- Expression
Mode = Most frequently occurring observation
- Name
Mode (Grouped Data)
- Note
L = lower boundary of modal class, f₁ = frequency of modal class, f₀ = frequency of class preceding modal class, f₂ = frequency of class succeeding modal class, h = class width.
- Expression
Mode = L + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] * h
- Name
Empirical Relationship between Mean, Median, Mode
- Note
Applicable for moderately skewed distributions.
- Expression
Mode = 3 Median - 2 Mean
Prerequisites
Basic arithmetic operations (addition, division, multiplication).
Understanding of data types (quantitative, qualitative, nominal, ordinal, interval, ratio).
Concept of frequency distribution and cumulative frequency.
Common mistakes
Not arranging data in order before calculating the Median.
Confusing formulas for ungrouped vs. grouped data.
Incorrectly identifying the modal class or median class in grouped data.
Ignoring the impact of outliers when choosing an appropriate measure.
Misapplying the empirical relationship between Mean, Median, and Mode.
Keywords
Statistics
Central Tendency
Mean
Median
Mode
Average
Frequency Distribution
Skewness
Outliers
Data Analysis
Practice preview
Which measure of central tendency is least affected by extreme values or outliers in a dataset?…
easy
A dataset has the following frequency distribution: | Class Interval | Frequency | |----------------|-----------| | 0-10 | 2 | | 10-20 | 4 | | 20-30 | 3 | | 30-40 …
medium
What is the primary purpose of measures of central tendency in statistics?…
easy