CBSE Class 7 Maths: Data Handling - Exercise 3.2 (Mode & Median)

Understanding the 'Mode' of Data

Have you ever wondered which observation in a dataset appears most frequently? That's exactly what the Mode helps us find! In simple words, the mode is the observation or value that occurs the highest number of times in a given set of data. Think of it as the "most popular" item.

For example, if your teacher asks students their favorite colours and the responses are: Red, Blue, Green, Red, Yellow, Blue, Red, Red. Here, Red appears 4 times, Blue appears 2 times, Green and Yellow appear 1 time each. Since Red appears most frequently, the Mode of this data set is 'Red'.

It's important to remember that a dataset can have:

1. One mode (unimodal): Like in our colour example, only 'Red' was the most frequent.
2. More than one mode (multimodal): If two or more values appear with the same highest frequency, then all of them are considered modes. For instance, if in another class, 'Red' appeared 3 times and 'Blue' also appeared 3 times, both Red and Blue would be modes.
3. No mode: If all observations occur the same number of times (e.g., each observation appears only once), then there is no mode for that dataset. For example, in the data set {1, 2, 3, 4, 5}, each number appears only once, so there is no mode.

Understanding the 'Median' of Data

While the mode tells us the most frequent value, the Median gives us the "middle" value of a dataset. Imagine arranging all your friends by height from shortest to tallest. The person exactly in the middle would represent the median height. For the median to be meaningful, it's crucial to arrange your data first, either in ascending order (smallest to largest) or descending order (largest to smallest).

Let's look at how to find the median:

Case 1: When the number of observations (n) is ODD.
If you have an odd number of observations, once you arrange them in order, the median is simply the value right in the middle. For example, consider the marks of 5 students: 15, 12, 18, 10, 20. First, arrange them in ascending order: 10, 12, 15, 18, 20. Here, there are 5 observations. The middle value is the 3rd one, which is 15. So, the median is 15.

Case 2: When the number of observations (n) is EVEN.
If you have an even number of observations, there isn't a single "middle" value. Instead, there will be two middle values. In this case, the median is the average (mean) of these two middle values. For example, consider the ages of 6 students: 11, 13, 10, 12, 14, 15. Arrange them: 10, 11, 12, 13, 14, 15. The two middle values are 12 and 13. To find the median, we calculate their average: (12 + 13) / 2 = 25 / 2 = 12.5. So, the median is 12.5.

The median is often a good measure of central tendency because it's not affected by extremely high or low values (outliers) as much as the mean can be.

Step-by-Step: Finding Mode and Median

1. Step 1: Arrange the Data — Always start by arranging the given data in either ascending (smallest to largest) or descending (largest to smallest) order. This step is crucial for finding the median and can also make finding the mode easier. Let's take an example dataset: 13, 16, 12, 14, 19, 12, 14, 13, 14.
2. Step 2: Count Frequencies for Each Observation — After arranging (or even before), count how many times each distinct observation appears in the dataset. This helps in identifying the mode. Arranged data: 12, 12, 13, 13, 14, 14, 14, 16, 19 Frequencies: 12: 2 times 13: 2 times 14: 3 times 16: 1 time 19: 1 time
3. Step 3: Identify the Mode — Look for the observation(s) with the highest frequency. In our example, the number 14 appears 3 times, which is more than any other number. Therefore, the Mode is 14.
4. Step 4: Identify the Median — Now, use the arranged data to find the median. Count the total number of observations (n). In our example, there are 9 observations (n=9). If n is ODD: The median is the observation at the position (n+1)/2. For our example, (9+1)/2 = 10/2 = 5th position. Looking at the arranged data (12, 12, 13, 13, 14, 14, 14, 16, 19), the 5th observation is 14. Therefore, the Median is 14. If n is EVEN: The median is the average of the two middle observations at positions n/2 and (n/2) + 1.

Important Points to Remember for Mode and Median

• Always arrange data for Median: Forgetting to arrange the data in ascending or descending order is the most common mistake when finding the median. Without proper arrangement, you won't find the true middle value.
• Mode is about Frequency: The mode is determined solely by how often an observation appears, not its numerical value or position. It's the 'most common' value.
• Median is a Positional Average: The median is a value that divides the data into two equal halves when arranged. Half the observations are below it, and half are above it.
• Mean, Mode, and Median are Measures of Central Tendency: They all tell us something about the 'center' or 'typical' value of a dataset, but in different ways.
• Handling Multiple Modes: If two or more values have the same highest frequency, then all of them are modes for that dataset. It's not limited to just one.

Practice Questions with Solutions

• Q: The scores in mathematics test (out of 25) of 15 students are as follows: 19, 25, 23, 20, 9, 20, 15, 10, 5, 16, 25, 20, 24, 12, 20 Find the mode and median of this data. A: Step 1: Arrange the data in ascending order: 5, 9, 10, 12, 15, 16, 19, 20, 20, 20, 20, 23, 24, 25, 25 Step 2: Count the frequency of each score: 5 (1), 9 (1), 10 (1), 12 (1), 15 (1), 16 (1), 19 (1), 20 (4), 23 (1), 24 (1), 25 (2) Step 3: Identify the Mode. The score 20 appears most frequently (4 times). Mode = 20 Step 4: Identify the Median. There are 15 observations (n=15), which is an odd number. The median will be the observation at the (n+1)/2 position. (15+1)/2 = 16/2 = 8th position. From the arranged data, the 8th observation is 20. Median = 20 Final answer: Mode = 20, Median = 20
• Q: Following are the runs scored by a team in 6 matches: 23, 10, 39, 12, 42, 10 Find the mode and median of the runs scored. A: Step 1: Arrange the data in ascending order: 10, 10, 12, 23, 39, 42 Step 2: Count frequencies for each score: 10 (2), 12 (1), 23 (1), 39 (1), 42 (1) Step 3: Identify the Mode. The run score 10 appears most frequently (2 times). Mode = 10 Step 4: Identify the Median. There are 6 observations (n=6), which is an even number. The median will be the average of the two middle observations at n/2 and (n/2)+1 positions. n/2 = 6/2 = 3rd position. (n/2)+1 = 3+1 = 4th position. The 3rd observation is 12 and the 4th observation is 23. Median = (12 + 23) / 2 = 35 / 2 = 17.5 Final answer: Mode = 10, Median = 17.5
• Q: A survey of 10 families showed the number of children they have: 2, 3, 1, 2, 4, 2, 1, 3, 2, 5 Find the mode of the number of children. A: Step 1: Arrange the data in ascending order: 1, 1, 2, 2, 2, 2, 3, 3, 4, 5 Step 2: Count the frequency of each number of children: 1 (2 times), 2 (4 times), 3 (2 times), 4 (1 time), 5 (1 time) Step 3: Identify the Mode. The number '2' appears most frequently (4 times). Mode = 2 Final answer: The mode of the number of children is 2.
• Q: Find the median of the following dataset: 8, 12, 6, 10, 14, 9, 7 A: Step 1: Arrange the data in ascending order: 6, 7, 8, 9, 10, 12, 14 Step 2: Count the number of observations. There are 7 observations (n=7), which is an odd number. Step 3: Identify the Median. The median will be the observation at the (n+1)/2 position. (7+1)/2 = 8/2 = 4th position. From the arranged data, the 4th observation is 9. Median = 9 Final answer: The median of the dataset is 9.

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