Problem 61 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the mean of \(4,7,2,8,6\) and a is 7 , then the mean deviation from the median of these observations is [Online May 12, 2012] (a) 8 (b) 5 (c) 1 (d) 3

Step-by-Step Solution

Answer

3 (Option d)

Step 1: Calculate the Mean with Unknown "a"

The mean of the numbers \(4, 7, 2, 8, 6, a\) is 7. To find the value of \(a\), set up the equation: \[ \frac{4 + 7 + 2 + 8 + 6 + a}{6} = 7 \]. Solve it step by step: \(4 + 7 + 2 + 8 + 6 + a = 42\). Simplifying, \(27 + a = 42\). Thus \(a = 15\).

Step 2: Find the Median of the Data Set

Now that we have the complete data set \(4, 7, 2, 8, 6, 15\), arrange it in increasing order: \(2, 4, 6, 7, 8, 15\). The median of six numbers is the average of the third and fourth numbers: \(\frac{6 + 7}{2} = 6.5\).

Step 3: Calculate Deviations from the Median

Find the absolute deviation of each number from the median, \(6.5\): \(|2 - 6.5| = 4.5\), \(|4 - 6.5| = 2.5\), \(|6 - 6.5| = 0.5\), \(|7 - 6.5| = 0.5\), \(|8 - 6.5| = 1.5\), \(|15 - 6.5| = 8.5\).

Step 4: Calculate the Mean Deviation from the Median

Calculate the mean deviation using the formula: \( \frac{4.5 + 2.5 + 0.5 + 0.5 + 1.5 + 8.5}{6} \). The sum of deviations is \(18\), so the mean deviation is \(\frac{18}{6} = 3\).

Key Concepts

Median

The median is a measure of central tendency that reflects the middle value of a data set. It is particularly useful because it is less affected by extreme values than the mean.
For the data set in our example, consisting of the numbers 4, 7, 2, 8, 6, and 15, we must first arrange the values in increasing order: 2, 4, 6, 7, 8, 15.

• In a set with an odd number of values, the median is the middle number. However, this set has an even number of values (six).
• Therefore, the median is the average of the third and fourth numbers.
• In this case, the median is \[\frac{6 + 7}{2} = 6.5\]

Understanding how to find the median is important for accurately determining the central point, especially when calculating deviations.

Arithmetic Mean

The arithmetic mean, commonly referred to as the average, is another measure of central tendency. It provides an indication of the overall level of a data set.
To find the arithmetic mean, you sum up all the values and divide by the total number of values.
To illustrate, let's calculate the arithmetic mean when an unknown variable \(a\) is part of the data set 4, 7, 2, 8, 6, and \(a\). Given that the mean is 7, we set up the following equation:\[\frac{4 + 7 + 2 + 8 + 6 + a}{6} = 7\]Solving this equation gives us:

• \(4 + 7 + 2 + 8 + 6 + a = 42\)
• This simplifies to \(27 + a = 42\)
• Therefore, \(a = 15\)

The arithmetic mean tells us about the average value of the set and is useful in further calculations, such as finding deviations.

Absolute Deviation

Absolute deviation measures how much each data point differs from a central value, such as the median. It's useful for understanding the spread of data values around this center.
The process involves the following steps:

• Calculate the absolute deviation for each data point by subtracting the median and taking the absolute value. For example, the absolute deviation from the median (6.5 in our problem) is calculated as:
  • |2 - 6.5| = 4.5
  • |4 - 6.5| = 2.5
  • |6 - 6.5| = 0.5
  • |7 - 6.5| = 0.5
  • |8 - 6.5| = 1.5
  • |15 - 6.5| = 8.5
• To find the mean deviation, sum these absolute deviations and divide by the number of observations:
  \[\frac{4.5 + 2.5 + 0.5 + 0.5 + 1.5 + 8.5}{6} = 3\]

The mean of the absolute deviations, known as the mean deviation, gives insight into the variability of the data points relative to the median.