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Data Handling - Representative Values (Mean, Median, Mode)

Grade 7CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Data and Observations: Data is a collection of numerical facts gathered to provide information. Each individual value in the data set is called an observation. You can visualize data as a set of points scattered along a number line.

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Arithmetic Mean (Average): The mean is the 'average' value that represents the entire data set. Visually, if you think of data points as blocks of different heights, the mean is the height they would all have if you redistributed the blocks to make them level.

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Range: The range indicates the spread of the data. It is the difference between the highest and lowest values. On a horizontal scale, the range is the total length covered from the leftmost point to the rightmost point.

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Mode: The mode is the value that occurs most frequently in a data set. In a bar graph, the mode is easily identified as the value corresponding to the tallest bar. A data set may have one mode, more than one mode, or no mode at all.

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Median: The median is the middle-most value when the data is arranged in ascending or descending order. It divides the data set into two equal halves. Visually, if you line up all observations in a row by size, the median is the value sitting exactly in the center.

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Representative Values: Mean, Median, and Mode are collectively known as measures of central tendency. They provide a single value that describes the center or the 'typical' value of the data distribution.

📐Formulae

Arithmetic Mean=Sum of all observationsNumber of observations\text{Arithmetic Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}}

Range=Highest Observation−Lowest Observation\text{Range} = \text{Highest Observation} - \text{Lowest Observation}

Median (for n odd observations)=Value of (n+12)th observation\text{Median (for } n \text{ odd observations)} = \text{Value of } \left(\frac{n+1}{2}\right)^{\text{th}} \text{ observation}

💡Examples

Problem 1:

The marks obtained by 5 students in a math test are: 35,40,35,50,4535, 40, 35, 50, 45. Find the Mean and Range of this data.

Solution:

Step 1: To find the Mean, sum all the observations: 35+40+35+50+45=20535 + 40 + 35 + 50 + 45 = 205. \nStep 2: Divide the sum by the number of students (n=5n = 5): Mean=2055=41\text{Mean} = \frac{205}{5} = 41. \nStep 3: To find the Range, identify the highest and lowest values: Highest=50\text{Highest} = 50, Lowest=35\text{Lowest} = 35. \nStep 4: Range=50−35=15\text{Range} = 50 - 35 = 15.

Explanation:

The mean provides the average score of the group, while the range shows the performance gap between the highest and lowest scorer.

Problem 2:

Find the Mode and Median of the following data: 13,16,12,14,13,12,1313, 16, 12, 14, 13, 12, 13.

Solution:

Step 1: Arrange the data in ascending order: 12,12,13,13,13,14,1612, 12, 13, 13, 13, 14, 16. \nStep 2: Identify the most frequent value. 1313 appears three times, while others appear less. So, Mode=13\text{Mode} = 13. \nStep 3: Count the number of observations (n=7n = 7). Since nn is odd, the Median is the (7+12)th\left(\frac{7+1}{2}\right)^{\text{th}} observation. \nStep 4: Median is the 4th4^{\text{th}} term in the sorted list. In 12,12,13,13,13,14,1612, 12, 13, 13, 13, 14, 16, the 4th4^{\text{th}} term is 1313. So, Median=13\text{Median} = 13.

Explanation:

The mode is found by frequency counting, and the median is found by locating the center of the ordered list. In this specific case, both representative values are the same.