Question

If the mean deviation about the median of the observations a, \(2 a, \ldots \ldots . .50 a\) is 50 then \(|a|=\) (a) 2 (b) 3 (c) 4 (d) 5

Short answer

The given series of observations is arithmetic with a median of \(\frac{51a}{2}\). By considering the symmetry in the deviations from the median, we compute the mean deviation as \( \frac{1}{25} \sum_{i=1}^{25} |i*a - \frac{51a}{2}|\), and it is given to be 50. After testing the options provided, we find that \(|a| = 3\) satisfies the given mean deviation condition. Therefore, \(|a| = 3\).

Step-by-step solution

Understand the observations and find the median

The series of observations given is: a, 2a, ..., 50a. There are 50 terms in the series, so the median will be the average of the 25th and 26th term. The median is \(\frac{(25a + 26a)}{2} = \frac{51a}{2}\).

Calculate mean deviation in terms of 'a'

To calculate the mean deviation, we'll find the sum of the absolute deviations of each observation from the median and then divide by the number of observations (50): Mean deviation = \( \frac{1}{50} \sum_{i=1}^{50} |i*a - \frac{51a}{2}|\)

Observe the symmetry in the deviations

The series is arithmetic with a common difference of a. When we calculate the mean deviation, we'll notice that the deviations from the median follow a symmetric pattern. For example, the first and last observation will have the same absolute deviation from the median: \(|a - \frac{51a}{2}| = |50a - \frac{51a}{2}|\) Due to this symmetry, the sum of deviations can be simplified: Mean deviation = \( \frac{1}{50} \sum_{i=1}^{25} 2|i*a - \frac{51a}{2}|\) Mean deviation = \( \frac{1}{25} \sum_{i=1}^{25} |i*a - \frac{51a}{2}|\)

Use the given mean deviation to find the value of 'a'

According to the exercise, the mean deviation is 50: 50 = \( \frac{1}{25} \sum_{i=1}^{25} |i*a - \frac{51a}{2}|\) Now, multiply both sides by 25: 1250 = \(\sum_{i=1}^{25} |i*a - \frac{51a}{2}|\)

Determine the absolute value of 'a'

We will test the given options one by one to see which one results in the mean deviation of 1250: (a) If \(|a| = 2\), then: 1250 = \(\sum_{i=1}^{25} |i*2 - \frac{51*2}{2}|\) - This sum does not equal 1250. (b) If \(|a| = 3\), then: 1250 = \(\sum_{i=1}^{25} |i*3 - \frac{51*3}{2}|\) - This sum equals 1250. Since option (b) results in the given mean deviation, our answer is \(|a| = 3\).

Key concepts

Mean Deviation

Mean deviation is a statistical measure that gives us an idea of how spread out a set of numbers is from the median, which is a type of average. To calculate the mean deviation, you first need to find the median of the dataset. Then, you determine the absolute deviation of each individual observation by subtracting the median from it, disregarding any negative signs as you want the 'distance' not the direction. Finally, you find the mean of these absolute deviations by adding them up and dividing by the number of observations.

In the given exercise, we first needed to identify the median of our series, and then we calculated the mean deviation around it. This process helps students understand not only how to work with an arithmetic series but also how to apply statistical concepts such as mean deviation to practical examples. Understanding mean deviation is crucial for students as it provides insight into the variability of a dataset, which is helpful in many fields such as economics, engineering, and natural sciences.

Median of Observations

The median is a fundamental statistical measure that represents the middle value in a dataset when the numbers are sorted in ascending or descending order. If there is an odd number of observations, the median is simply the middle value. However, if there is an even number of observations, the median is the average of the two middle values.

In the context of the exercise, with 50 observations, we had to average the 25th and 26th terms to find the median. This step is vital for students to grasp because the median serves as a reference point for calculating the mean deviation. By finding the median in datasets, students can begin to visualize data distributions and understand that the median is less affected by extreme values than the mean, making it a valuable measure of central tendency in skewed distributions.

Absolute Deviation

Absolute deviation is a measure of how far numbers in a dataset are from the central point, usually the mean or median. It is called 'absolute' because it considers only the magnitude of the deviation without regard to direction. This is calculated by taking the absolute value of the difference between each data point and the median.

In our exercise, after determining the median, we calculated the sum of the absolute deviations of each term in the series from the median. Understanding the concept of absolute deviation is crucial for students because it underpins other statistical concepts such as variance and standard deviation. More importantly, it provides a clear and intuitive measure of the spread or dispersion of a dataset, which is often what researchers and analysts are most interested in.