How to Calculate Median and Mode for Grouped and Ungrouped Data
Understanding Statistical Measures of Centre Median and Mode Formulas is essential for every student of mathematics. These concepts not only form the foundation of statistics for school and board exams, but they are also frequently used in competitive exams and real-life data analysis. By mastering the median and mode formulas, you gain the ability to summarize and interpret data more effectively.
Core Concept: Measuring the Centre of Data
The statistical measures of centre, or measures of central tendency, help us find a single number that represents the entire data set. Among these, the median and mode are two crucial concepts in statistics. The median locates the middle value, while the mode identifies the value that occurs most frequently. These measures help interpret grouped and ungrouped data in subjects ranging from social science to economics. At Vedantu, we make these concepts simple and exam-ready for you!
Median and Mode: Definitions & Importance
- Median: The centre value of a data set when the numbers are arranged in order. If there are an even number of items, the median is the mean of the two middle values.
- Mode: The value that appears most often in a data set. There can be more than one mode, or sometimes none if all numbers occur equally.
For example, in the data set [4, 6, 7, 7, 8], the median is 7 and the mode is also 7. These measures are important because they are less affected by outliers or extreme values, making them great for summarizing skewed data sets.
Formulas for Median and Mode
Median and mode have different formulas depending on whether the data is grouped in classes (continuous data) or provided as a simple list (ungrouped data). Understanding both is vital for exams and for practical analysis.
| Type of Data | Median Formula | Mode Formula |
|---|---|---|
| Ungrouped Data |
|
The value that occurs most frequently in the data set. |
| Grouped Data |
Median = l + [(\( \frac{N}{2} \) – cf)/f] × h Where: l = lower class boundary of median class, N = total frequency, cf = cumulative frequency before median class, f = frequency of median class, h = class width. |
Mode = l + [(\( f_m – f_1 \))/(\( 2f_m – f_1 – f_2 \))] × h, Where: l = lower boundary of modal class, \( f_m \) = frequency of modal class, \( f_1 \) = frequency of class before modal, \( f_2 \) = frequency of class after modal, h = class width. |
Worked Examples
Example 1: Median in Ungrouped Data
Find the median of 8, 10, 12, 14, 10.
- Arrange the data: 8, 10, 10, 12, 14
- n = 5 (odd), so median is value at position (5+1)/2 = 3rd: 10
Example 2: Median in Grouped Data
Class intervals and frequency:
10-20: 3
20-30: 6
30-40: 7
40-50: 4
- Total frequency (N) = 3+6+7+4 = 20
- N/2 = 10, cumulative frequencies: 3, 9, 16, 20
- Median class is 30-40 (cf just reaches/exceeds 10)
- l = 30, cf before median class = 9, f = 7, h = 10
- Median = 30 + [(10 – 9)/7] × 10 = 30 + 1.43 ≈ 31.43
Example 3: Mode in Grouped Data
Using the frequencies above:
Modal class is 30-40 (frequency 7), f₁ = 6 (class before), f₂ = 4 (class after), h = 10, l = 30.
Mode = 30 + [(7 – 6)/(2×7 – 6 – 4)] × 10 = 30 + (1/4)×10 = 32.5
Practice Problems
- Find the median and mode of the data: 5, 7, 7, 8, 10, 10, 12
- Calculate the median for the classes: 0-10: 2, 10-20: 5, 20-30: 8, 30-40: 5
- Which is more affected by outliers: mean, median, or mode?
- Construct a grouped frequency table with a given data set and find its mode.
- In a data set, mean = 15, median = 18. Find mode using the empirical relation.
Common Mistakes to Avoid
- Not arranging ungrouped data in increasing order before finding the median.
- Mistaking frequency for cumulative frequency in median formula for grouped data.
- Forgetting to use class boundaries (not class limits) in formulas for grouped data.
- Assuming data always has a unique mode – some data sets can be bimodal or have no mode.
Real-World Applications
The concepts of median and mode are widely used in daily life. For example, medians are used to report household incomes or land prices, as they are not affected by a few extremely high or low values. Mode is often used in business to find the most popular product size or in surveys to find the most common response. At Vedantu, we connect these classroom concepts to real-world uses, making learning practical and meaningful.
For more related topics, check out Mean, Variance, or learn the Difference Between Mean, Median and Mode.
In this lesson, we have explored Statistical Measures of Centre Median and Mode Formulas, their definitions, formulas, and worked examples. Mastering these concepts will empower you to summarize and interpret data more confidently in both academics and real-life scenarios. With Vedantu, statistics becomes easier and more fun to learn!
FAQs on Statistical Measures of Centre: Median and Mode Formulas
1. What is the formula of median for grouped and ungrouped data?
The median is the middle value in a dataset. For ungrouped data, arrange values in ascending order; if the number of values (n) is odd, the median is the value at the (n+1)/2 position; if n is even, it's the average of values at n/2 and (n/2)+1 positions. For grouped data, the formula is: Median = l + [(N/2 – cf)/f] × h, where l = lower boundary of median class, N = total frequency, cf = cumulative frequency before median class, f = frequency of median class, and h = class interval width.
2. What is the formula of mode?
The mode is the most frequent value. For ungrouped data, it's the value appearing most often. For grouped data, the formula is: Mode = l + [(fm–f1)/(2fm–f1–f2)] × h, where l = lower boundary of modal class, fm = frequency of modal class, f1 = frequency of the class before the modal class, f2 = frequency of the class after the modal class, and h = class interval width.
3. How do you determine the median in a data set?
To find the median, first arrange your data in ascending order. If you have an odd number of data points, the median is the middle value. If you have an even number, the median is the average of the two middle values. For grouped data, use the formula: Median = l + [(N/2 – cf)/f] × h.
4. When do you use median vs. mode?
The median is best for skewed data or data with outliers, as it's less sensitive to extreme values. The mode is useful for finding the most common value in a dataset, particularly for categorical data (e.g., favourite colours). Both are measures of central tendency alongside the mean.
5. What is the empirical relation between mean, median, and mode?
For a moderately skewed distribution, the empirical relationship between the mean, median, and mode is: Mode = 3 × Median – 2 × Mean. This is helpful for estimating one measure if you know the other two. This formula applies to moderately skewed distributions, not highly irregular ones.
6. What is the median of 1,2,3,4,5,6,7,8,9,10?
The median is the middle value. Since there's an even number of values (10), we take the average of the two middle numbers (5 and 6). Therefore, the median is (5+6)/2 = 5.5.
7. What is the median of 13, 16, 12, 14, 19, 12, 14, 13, 14?
First, arrange the numbers in ascending order: 12, 12, 13, 13, 14, 14, 14, 16, 19. There are 9 numbers (odd), so the median is the middle value, which is the 5th number: 14.
8. How to calculate the mean, median, and mode of statistical data?
To calculate the mean, add all values and divide by the total number of values. The median is the middle value (or the average of the two middle values for even datasets) after arranging in order. The mode is the most frequent value. For grouped data, specific formulas are used for median and mode (see above).
9. What is the formula of mean, median mode class 10?
The mean is the sum of all values divided by the number of values. The median and mode formulas for class 10 depend on whether the data is grouped or ungrouped (see formulas above). Understanding these measures of central tendency is crucial for your class 10 exams.
10. Mean, median mode formula for grouped data?
For grouped data, the mean is calculated using the midpoints of the class intervals and their frequencies. The median and mode are calculated using the formulas provided earlier for grouped data. Remember that the class width and cumulative frequency play a crucial role in calculations.
11. Why can a data set have more than one mode?
A dataset can have more than one mode (bimodal or multimodal) if two or more values share the highest frequency. This indicates that there is more than one most common value within the dataset.
12. How do outliers affect the mode and median?
Outliers (extreme values) have little to no effect on the mode. The median is also relatively unaffected by outliers because the median is based on the position of values within the ordered dataset, rather than the actual values themselves.
