How to Calculate Variance and Standard Deviation with Step-by-Step Examples
The Mean Absolute Deviation Formula is a fundamental topic in statistics that helps students understand how much data points in a set vary from their average value. This concept is crucial for school-level mathematics, board exams like CBSE/ICSE, Olympiads, and competitive exams such as JEE, as well as for analyzing data in real-world situations involving business, science, or technology.
What is Mean Absolute Deviation?
Mean Absolute Deviation (MAD) is a measure of statistical dispersion. It calculates the average of the absolute differences between each data value and the mean (average) of the dataset. In simple words, it tells us, on average, how far each data point is from the mean. This makes it a handy tool to check for the variability or consistency of data and is less affected by extreme values compared to standard deviation. At Vedantu, we make learning such statistical concepts easy and student-friendly.
Mean Absolute Deviation Formula
The Mean Absolute Deviation Formula is used for both grouped and ungrouped data and represents the mean of the absolute differences from the dataset’s mean.
| Type of Data | Formula | Explanation of Terms |
|---|---|---|
| Ungrouped Data | \( MAD = \frac{\sum |x_{i} - \mu|}{n} \) |
|
| Grouped Data | \( MAD = \frac{\sum f_{i}|x_{i} - \bar{x}|}{\sum f_{i}} \) |
|
This formula helps in understanding how clustered or spread the data points are in a dataset. For a more detailed explanation of variance and dispersion, you can visit the Variance and Dispersion pages on Vedantu.
Step-by-Step Worked Examples
Example 1: Ungrouped Data
Find the Mean Absolute Deviation for this dataset: 12, 15, 18, 20, 23
- Calculate the mean: \( (12 + 15 + 18 + 20 + 23) / 5 = 88 / 5 = 17.6 \)
- Find the absolute deviation from the mean for each value:
- |12 - 17.6| = 5.6
- |15 - 17.6| = 2.6
- |18 - 17.6| = 0.4
- |20 - 17.6| = 2.4
- |23 - 17.6| = 5.4
- Add the absolute deviations: 5.6 + 2.6 + 0.4 + 2.4 + 5.4 = 16.4
- Divide by the number of data points:
- MAD = 16.4 / 5 = 3.28
So, the MAD for the set is 3.28.
Example 2: Grouped Data
Consider this frequency table:
| Class Interval | Frequency (\(f_i\)) |
|---|---|
| 10-20 | 5 |
| 20-30 | 8 |
| 30-40 | 12 |
| 40-50 | 10 |
- Find the mid-point of each class: 15, 25, 35, 45
- Calculate mean (\(\bar{x}\)):
- Sum = (5×15) + (8×25) + (12×35) + (10×45) = 75 + 200 + 420 + 450 = 1145
- Total frequency = 5 + 8 + 12 + 10 = 35
- Mean = 1145 / 35 ≈ 32.71
- Find absolute deviation for each class midpoint from the mean:
- |15 - 32.71| = 17.71
- |25 - 32.71| = 7.71
- |35 - 32.71| = 2.29
- |45 - 32.71| = 12.29
- Multiply each deviation by class frequency and sum:
- 5×17.71 = 88.55
- 8×7.71 = 61.68
- 12×2.29 = 27.48
- 10×12.29 = 122.90
- Total = 88.55 + 61.68 + 27.48 + 122.90 = 300.61
- Divide by total frequency: 300.61 / 35 ≈ 8.59
Therefore, MAD for the grouped data is approximately 8.59.
Practice Problems
- Find the MAD for the data set: 6, 8, 10, 12.
- Given the data set: 25, 24, 26, 23, 22, calculate its MAD.
- For the frequencies:
Score: 2, 4, 6, Frequency: 3, 5, 2
Find the MAD of the series. - In a class, the marks obtained by students in a test are: 15, 18, 17, 20, 15, 19. Find the MAD.
- Construct a grouped frequency table and compute MAD for data: 10, 11, 12, 13, 14, 15, 16, grouped in intervals of 2.
Common Mistakes to Avoid
- Forgetting to take the absolute value of the deviations (missing the modulus sign).
- Confusing MAD with variance or standard deviation formulas.
- Omitting to divide by the correct total: number of data points (ungrouped), total frequency (grouped).
- Not using class midpoints for grouped data.
Real-World Applications
Mean Absolute Deviation is used to measure the consistency of products in manufacturing, evaluate the stability of test scores among students, and analyze stock market volatility. For example, a company can use MAD to see how much product weights vary from the average, ensuring quality control. In daily life, you might use MAD to compare variability in travel times to school over a week.
In summary, the Mean Absolute Deviation Formula is a simple yet essential statistical tool for quantifying data variation. Understanding MAD helps students interpret data sets, solve exam problems efficiently, and relate concepts to real-world scenarios. To master MAD and other statistical measures, explore more resources and live classes at Vedantu, where complex topics are simplified for easy learning.
For more practice and theory, check out these Vedantu resources: Mean Absolute Deviation, Variance, and Standard Deviation.
FAQs on Statistical Measures of Variation: Formulas, Concepts & Examples
1. What are the four measures of variation?
The four main measures of variation in statistics are: range, variance, standard deviation, and mean absolute deviation. These measures describe the spread or dispersion of data points around the central tendency (mean, median, mode).
2. What is the formula for variance?
The variance formula differs slightly for population and sample data. For a population, it's the average of the squared differences from the mean: σ² = Σ(xᵢ - μ)² / N. For a sample, it's: s² = Σ(xᵢ - x̄)² / (n - 1), where n-1 is used for a less biased estimate.
3. How do you calculate the standard deviation of a dataset?
Standard deviation (SD) measures the spread of data around the mean. First, calculate the variance (as above). Then, take the square root of the variance. For a population: σ = √σ². For a sample: s = √s².
4. What does variability mean in statistics?
Variability, or dispersion, describes how spread out a dataset is. High variability indicates data points are widely scattered; low variability means they're clustered closely together around the mean. Understanding variability is crucial for data analysis and interpretation.
5. When should I use variance vs. standard deviation?
Both variance and standard deviation measure data spread. However, standard deviation is often preferred because it's in the same units as the original data, making it easier to interpret. Variance is useful in certain statistical calculations but is less intuitive to understand directly.
6. What is the formula for measuring variance?
The formula for variance depends on whether you're working with a population or a sample. For a population, it is: σ² = Σ(xᵢ - μ)² / N. For a sample, it's: s² = Σ(xᵢ - x̄)² / (n - 1). Remember that 'μ' represents the population mean, 'x̄' the sample mean, N the population size, and 'n' the sample size.
7. What is the variance of the data set 2, 4, 4, 4, 5, 5, 7, 9?
To find the variance, first calculate the mean (average) of the dataset. Then, find the squared difference of each data point from the mean. Finally, average these squared differences. Because this is a sample, use the sample variance formula: s² = Σ(xᵢ - x̄)² / (n - 1). The result will be the sample variance.
8. What is the formula for calculating SD?
The formula for calculating the standard deviation (SD) is the square root of the variance. If you have the population variance (σ²), then the population standard deviation is σ = √σ². If you have the sample variance (s²), then the sample standard deviation is s = √s².
9. What are measures of variability?
Measures of variability (or dispersion) quantify the spread or scatter of data points in a dataset. They show how much the data values differ from each other and from the central tendency. Common measures include range, interquartile range, variance, and standard deviation.
10. How do outliers affect measures of variation?
Outliers, extreme values in a dataset, significantly impact measures of variation, especially the range and standard deviation. Outliers inflate these measures, making the variability appear greater than it might be if the outliers were removed or accounted for differently. The mean absolute deviation is less susceptible to outliers.
