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Data Handling - Mean, Median, and Mode of ungrouped 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 collected for a specific purpose. Each numerical entry is called an observation. Imagine a list of marks in a class; raw data is the messy list of marks as they were recorded, while organized data is when they are sorted into a table or sequence.

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Arithmetic Mean: The Mean is the average of a set of numbers. Visually, it represents the 'balance point' of the data. If you represented each value as a stack of blocks, the Mean would be the height of each stack if you redistributed all blocks to make the stacks equal.

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Median: The Median is the middle value of the data when it is arranged in ascending or descending order. Think of a line of students arranged by height; the person standing exactly in the center is the median student. If there is an even number of students, the median is the average of the two students in the middle.

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Mode: The Mode is the observation that occurs most frequently in the dataset. In a bar graph, the mode corresponds to the tallest bar. A dataset can have one mode, more than one mode (bimodal or multimodal), or no mode at all if all values appear once.

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Range: The Range is the difference between the highest and lowest values in the dataset. It describes the spread of the data. Visually, it is the distance between the leftmost and rightmost points on a number line representing the data.

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Array: An array is the systematic arrangement of raw data in either ascending (smallest to largest) or descending (largest to smallest) order. This is a mandatory first step for finding the median.

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Frequency: Frequency refers to the number of times a particular observation occurs in the data. Using tally marks in a frequency distribution table helps visualize how the data points are distributed.

📐Formulae

Arithmetic Mean=Sum of all observationsTotal number of observations=∑xn\text{Arithmetic Mean} = \frac{\text{Sum of all observations}}{\text{Total number of observations}} = \frac{\sum x}{n}

Range=Maximum Value−Minimum Value\text{Range} = \text{Maximum Value} - \text{Minimum Value}

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

💡Examples

Problem 1:

The marks obtained by 7 students in a math test are: 15,12,18,15,20,10,1515, 12, 18, 15, 20, 10, 15. Find the Mean, Median, and Mode.

Solution:

  1. Mean: Mean=15+12+18+15+20+10+157=1057=15\text{Mean} = \frac{15 + 12 + 18 + 15 + 20 + 10 + 15}{7} = \frac{105}{7} = 15
  2. Median: First, arrange in ascending order: 10,12,15,15,15,18,2010, 12, 15, 15, 15, 18, 20. Since n=7n = 7 (odd), the Median is the (7+12)th=4th\left(\frac{7+1}{2}\right)^{th} = 4^{th} term. The 4th4^{th} term is 1515.
  3. Mode: The value 1515 appears 3 times, which is more than any other value. So, Mode=15\text{Mode} = 15.

Explanation:

To solve this, we summed all values for the mean, sorted the list to find the middle position for the median, and identified the most frequent number for the mode.

Problem 2:

Find the range and the median of the following set of data: 25,30,22,18,35,40,28,3225, 30, 22, 18, 35, 40, 28, 32.

Solution:

  1. Range: Max=40,Min=18\text{Max} = 40, \text{Min} = 18. Range=40−18=22\text{Range} = 40 - 18 = 22.
  2. Median: Arrange in ascending order: 18,22,25,28,30,32,35,4018, 22, 25, 28, 30, 32, 35, 40. Here, n=8n = 8 (even). Middle terms are the (82)th=4th\left(\frac{8}{2}\right)^{th} = 4^{th} term and the (82+1)th=5th\left(\frac{8}{2} + 1\right)^{th} = 5^{th} term. 4th term=284^{th} \text{ term} = 28 and 5th term=305^{th} \text{ term} = 30. Median=28+302=582=29\text{Median} = \frac{28 + 30}{2} = \frac{58}{2} = 29.

Explanation:

Since the number of observations is even, the median is the arithmetic average of the two central terms after sorting.