Unit 3: Measures of central tendency
Fundamental of Statistics notes · PTU syllabus (UGCC2504)
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Unit summary
An average is a single value that represents a whole set of data. Averages let us compare groups quickly — the average marks of two sections, or the typical salary in two cities. This unit covers the arithmetic mean, median, mode and harmonic mean.
After this unit you can
- List the characteristics of a good average
- Calculate the mean, median and mode for individual, discrete and continuous series
- Explain the merits and demerits of each average
- Calculate the harmonic mean and know when to use it
PTU syllabus topics
- Purpose and characteristics of a good average
- arithmetic mean
- median
- mode — calculation
- merits and demerits of each
- harmonic mean and its properties
Arithmetic mean
x̄ = Σx / n
Weighted mean
x̄w = Σwx / Σw
Median (grouped)
L + [(n/2 − cf) / f] × h
Mode (grouped)
L + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h
Harmonic mean
n / Σ(1/x)
Topic 1
Characteristics of a good average
A good average should be: rigidly defined, easy to understand and calculate, based on all observations, not unduly affected by extreme values, capable of further algebraic treatment, and stable from sample to sample.
Topic 2
Arithmetic mean
The arithmetic mean is the sum of values divided by their number.
Individual series
x̄ = Σx / n
Discrete series
x̄ = Σfx / Σf
Continuous series
x̄ = Σfm / Σf
m = mid-value of each class
Short-cut method
x̄ = A + Σfd / Σf
d = m − A, A = assumed mean
Example
Marks 40, 50, 60, 70, 80: mean = 300 / 5 = 60.
Merits: easy, uses all values, algebraic treatment possible. Demerits: affected by extreme values, cannot be found for open-ended classes without assumptions.
Topic 3
Median
The median is the middle value when data is arranged in order. It divides the data into two equal halves.
- Individual series: median = value of the (n + 1)/2th item.
- Continuous series: median class contains the N/2th item; Median = L + [(N/2 − cf) / f] × h, where L = lower limit of the median class, cf = cumulative frequency before it, f = its frequency, h = class width.
Example
Data 3, 7, 9, 12, 15: n = 5, median = 3rd item = 9.
Merits: not affected by extreme values, suitable for open-ended classes. Demerits: not based on all observations, needs arranging.
Topic 4
Mode
The mode is the value that occurs most often.
- Continuous series: Mode = L + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h, where f₁ = frequency of the modal class, f₀ and f₂ = frequencies of the classes before and after.
- Empirical relation: Mode = 3 Median − 2 Mean (for moderately skewed data).
Merits: easy, unaffected by extremes, useful for the "most popular" size or brand. Demerits: may not exist or may not be unique, not based on all items.
Topic 5
Harmonic mean
The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals: HM = n / Σ(1/x). It is used to average rates and speeds where the distance or work is fixed.
Example
A car goes 60 km at 30 km/h and returns at 60 km/h. Average speed = HM = 2 / (1/30 + 1/60) = 40 km/h, not 45.
For positive values: AM ≥ GM ≥ HM.
Mean
Yes
Yes
Median
No
No
Mode
No
No
Key terms
- Average
- A single value representing a data set
- Arithmetic mean
- Sum of values divided by their count
- Median
- The middle value of ordered data
- Mode
- The most frequently occurring value
- Harmonic mean
- n divided by the sum of reciprocals; used for rates
Quick revision
- Mean = Σfx / Σf; median = L + [(N/2 − cf)/f]h; mode = L + [(f₁ − f₀)/(2f₁ − f₀ − f₂)]h.
- Mode = 3 Median − 2 Mean.
- HM for speeds and rates; AM ≥ GM ≥ HM.
- Median and mode are not affected by extreme values.
Important exam questions
Practice questions written to the PTU exam pattern for this unit's syllabus: short answers (Section A style) and long answers (Sections B and C style).
Short-answer questions
- Q1.State the characteristics of a good average.
- Q2.Find the median of 5, 9, 2, 7, 11, 4.
- Q3.Write the empirical relation between mean, median and mode.
- Q4.When is the harmonic mean used?
- Q5.Give two merits and two demerits of the arithmetic mean.
Long-answer questions
- Q1.Calculate the mean, median and mode of a continuous frequency distribution of your choice.
- Q2.Compare the mean, median and mode and explain which is the best average.
- Q3.Explain the harmonic mean with an example and show that AM ≥ HM for two numbers.
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