Elementary Statistics Explained with Basic Examples
Elementary Statistics deals with collecting, organising, summarising and interpreting numerical data. This chapter covers mean, median, mode, range, variance, standard deviation, frequency distributions and weighted averages. It also explains grouped-data formulas, positional measures and calculation methods used in elementary statistics aptitude questions.
On this page
What Is Elementary Statistics?
For observations x₁, x₂, …, xₙ, the arithmetic mean is x̄ = (x₁ + x₂ + … + xₙ)/n. The median is the middle value after arranging the data, while the mode is the value occurring most frequently. Measures such as range, variance and standard deviation describe the spread of the data.
Elementary Statistics Video Lessons
Watch short topic-wise lessons for quick revision.
Range, Variance and Standard Deviation
Learn how to calculate and interpret range, variance, and standard deviation as measures of dispersion in statistics, including the role of each measure in describing data spread.
Elementary Statistics Formula & Tricks
Important Formulas
Add all observations and divide the total by the number of observations.
Here, x is the value, f is its frequency and Σf is the total frequency.
Multiply each value by its weight, add the products and divide by the sum of weights.
Arrange the observations in ascending or descending order before locating the position.
l is the lower boundary of the median class, N is total frequency, c.f. is cumulative frequency before that class, f is its frequency and h is class width.
l is the lower boundary of the modal class, f₁ its frequency, f₀ the preceding frequency, f₂ the succeeding frequency and h the class width.
Range measures the difference between the maximum and minimum observations.
Variance is the mean of squared deviations from the mean, and standard deviation is its positive square root.
Quick Tricks
Choose a convenient assumed mean A and write d = x − A. Then x̄ = A + Σd/n. For a frequency distribution, x̄ = A + Σfd/Σf.
Adding or subtracting the same number from every observation changes the mean, median and mode by that number, but it does not change the range or standard deviation.
In a frequency table, calculate cumulative frequencies first. The median class is the first class whose cumulative frequency is at least N/2.
Elementary Statistics Concepts
Mean, Median and Mode
For n observations, mean = Σx/n. Arrange data before finding the median. For an even number of observations, average the two central values. A data set may have one mode, more than one mode or no mode if no value repeats. For a moderately skewed distribution, the empirical relation is Mode ≈ 3 Median − 2 Mean; this is an approximation, not an exact identity.
Frequency Distribution and Weighted Mean
For discrete data, multiply every value x by its frequency f. For class intervals, use the class mark x = (lower limit + upper limit)/2 before applying the formula. A weighted mean uses x̄w = Σwx/Σw when observations have unequal weights.
Median and Mode in Grouped Data
The median class contains the N/2th observation in the cumulative-frequency table. The modal class has the highest frequency. Using class boundaries is necessary when intervals are discontinuous, such as 10–19 and 20–29; their continuous boundaries are 9.5–19.5 and 19.5–29.5.
Measures of Dispersion
Range uses only the largest and smallest values. Variance uses squared deviations, so it cannot be negative. Standard deviation has the same unit as the original data and is zero only when all observations are equal. For population data, σ² = Σ(x − x̄)²/n.
Combined Mean and Data Properties
For groups with sizes n₁, n₂, … and means x̄₁, x̄₂, …, combined mean = (n₁x̄₁ + n₂x̄₂ + …)/(n₁ + n₂ + …). The mean is affected by every observation, whereas the median is less affected by extreme values. If every observation is multiplied by k, the mean, median, mode and standard deviation are multiplied by |k|, and the variance is multiplied by k².
Elementary Statistics Data Table
Use the structured data below while following the explanation and examples.
Comparison of Basic Statistical Measures
Definitions and formulas for common elementary statistics measures.
| Measure | Formula or rule | Main characteristic |
|---|---|---|
| Mean | Σx/n | Uses every observation |
| Median | Middle arranged value | Depends on position |
| Mode | Most frequent value | Depends on frequency |
| Range | Maximum − minimum | Uses only two extreme values |
| Variance | Σ(x − x̄)²/n | Squared spread from the mean |
| Standard deviation | √variance | Spread in the original unit |
Note: The variance formula shown is for population data. For a sample, sample variance is commonly written as s² = Σ(x − x̄)²/(n − 1).
Learn more concepts on PrepShots
Start ₹1 Trial →Practice Elementary Statistics Questions
Practise published questions related to this topic.
Want more practice questions?
Practice in App →Elementary Statistics Quick Quiz
Attempt 5 questions and check your score instantly.
Keep practising
Practice more Elementary Statistics questions in the PrepShots app and continue from your current topic.
Practice More Questions - Start ₹1 Trial →Quick Revision Notes
Elementary Statistics Revision Points
Key formulas and rules for quick revision.
- Mean = Σx/n; frequency mean = Σfx/Σf; weighted mean = Σwx/Σw.
- Arrange observations before finding the median.
- For even n, median is the average of the n/2th and (n/2 + 1)th observations.
- Grouped median = l + [(N/2 − c.f.)/f]h.
- Grouped mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)]h.
- Range = maximum − minimum.
- Population variance = Σ(x − x̄)²/n and standard deviation = √variance.
- Adding a constant changes mean, median and mode but not range or standard deviation; multiplying by k multiplies standard deviation by |k| and variance by k².
Elementary Statistics FAQs
What is the formula for the arithmetic mean?
The arithmetic mean is x̄ = Σx/n, where Σx is the sum of observations and n is their number. For 4, 6 and 8, the mean is 18/3 = 6.
How is the median found when the number of observations is even?
After arranging the data, median = average of the n/2th and (n/2 + 1)th observations. For 2, 5, 9 and 12, the median is (5 + 9)/2 = 7.
What is the mean of a frequency distribution?
Use x̄ = Σfx/Σf. For values 1 and 3 with frequencies 2 and 4, mean = (1×2 + 3×4)/6 = 14/6 = 7/3.
How do you identify the median class?
Calculate N/2 and cumulative frequencies. The first class whose cumulative frequency is at least N/2 is the median class.
What is the difference between variance and standard deviation?
Variance is the average of squared deviations from the mean: σ² = Σ(x − x̄)²/n. Standard deviation is the positive square root of variance and has the same unit as the data.
What happens to the mean if 5 is added to every observation?
The mean also increases by 5. The median and mode increase by 5, while the range, variance and standard deviation remain unchanged.