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Data Handling - Collection and Organisation of Data

Grade 7ICSE

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 or figures gathered to provide specific information. Each individual numerical entry in the data set is called an observation. For example, if we record the weights of 5 students as 35,40,38,35,4235, 40, 38, 35, 42 kg, each weight is an observation.

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Raw and Organised Data: Data collected in its original form is called raw data. To make it meaningful, we organize it into a Frequency Distribution Table. This table typically features three columns: the variable (observation), Tally Marks (visual strokes), and Frequency (the number of times an observation repeats).

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Tally Marks: This is a visual method of counting frequencies. For each occurrence of a value, a vertical stroke ∣| is placed. To represent five, a diagonal line is drawn across four vertical strokes, creating a bundle that looks like a barred gate. This grouping allows for quick visual totaling of data points.

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Arithmetic Mean: The Mean is the average of the given set of data. It is calculated by dividing the sum of all observations by the total number of observations. Visually, if you imagine each data point as a weight on a scale, the Mean is the 'balance point' where the scale stays level.

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Median: The Median is the value of the middle-most observation when the data is arranged in ascending or descending order. If the data set has an odd number of observations, the median is the exact middle term. If it has an even number, the median is the average of the two central terms.

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Mode: The Mode is the observation that occurs most frequently in a data set. In a frequency distribution table, it is the value corresponding to the highest frequency. A data set can have more than one mode (bimodal) or no mode at all if all values appear only once.

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Range: The Range is the measure of the spread of the data. It is calculated as the difference between the maximum value and the minimum value in the data set. A larger range indicates that the data points are more spread out from each other.

📐Formulae

Arithmetic Mean=Sum of all observationsTotal number of observations\text{Arithmetic Mean} = \frac{\text{Sum of all observations}}{\text{Total number of observations}}

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

Median (if n is odd)=(n+12)th observation\text{Median (if } n \text{ is odd)} = \left(\frac{n+1}{2}\right)^{\text{th}} \text{ observation}

Median (if n is even)=(n2)th term+(n2+1)th term2\text{Median (if } n \text{ is even)} = \frac{\left(\frac{n}{2}\right)^{\text{th}} \text{ term} + \left(\frac{n}{2} + 1\right)^{\text{th}} \text{ term}}{2}

💡Examples

Problem 1:

The marks obtained by 7 students in a math test are: 15,20,15,18,25,15,2215, 20, 15, 18, 25, 15, 22. Find the Mean, Median, and Mode.

Solution:

Step 1: Calculate Mean: Mean=15+20+15+18+25+15+227=1307≈18.57\text{Mean} = \frac{15+20+15+18+25+15+22}{7} = \frac{130}{7} \approx 18.57. \nStep 2: Arrange data in ascending order: 15,15,15,18,20,22,2515, 15, 15, 18, 20, 22, 25. \nStep 3: Find Median: Since n=7n=7 (odd), Median=(7+12)th term=4th term=18\text{Median} = \left(\frac{7+1}{2}\right)^{\text{th}} \text{ term} = 4^{\text{th}} \text{ term} = 18. \nStep 4: Find Mode: The value 1515 appears 3 times, which is the highest frequency. So, Mode=15\text{Mode} = 15.

Explanation:

To find central tendencies, we first sum the values for the mean, sort them to find the middle value for the median, and identify the most frequent value for the mode.

Problem 2:

Find the Range and Median of the following data: 12,8,15,20,10,1812, 8, 15, 20, 10, 18.

Solution:

Step 1: Identify Highest and Lowest values: Highest=20,Lowest=8\text{Highest} = 20, \text{Lowest} = 8. \nStep 2: Calculate Range: Range=20−8=12\text{Range} = 20 - 8 = 12. \nStep 3: Arrange in ascending order: 8,10,12,15,18,208, 10, 12, 15, 18, 20. \nStep 4: Find Median: Since n=6n=6 (even), Median=3rd term+4th term2=12+152=272=13.5\text{Median} = \frac{3^{\text{rd}} \text{ term} + 4^{\text{th}} \text{ term}}{2} = \frac{12 + 15}{2} = \frac{27}{2} = 13.5.

Explanation:

The range shows the spread between the extremes. For the median of an even-numbered data set, we take the average of the two middle numbers after sorting.