Sum of Absolute Deviations from Central Tendency Measures
This question asks us to identify the measure of central tendency from which the sum of the absolute deviations of the data points is the smallest. In statistics, "ignoring signs" means we are considering the absolute value of the deviations.
Understanding Deviations from the Median
A fundamental property related to the Median is that it uniquely minimizes the sum of the absolute deviations for a given dataset. This means if you calculate the difference between each data point and the Median, take the absolute value of each difference, and then sum them up, you will get the smallest possible sum compared to using any other single value from the dataset or outside of it.
For a dataset denoted as {$x_1, x_2, ..., x_n$}, the value $M$ that minimizes the sum $S = \sum_{i=1}^{n} |x_i - M|$ is the Median of the dataset.
Comparing with Other Measures
Let's look at why the other options are not correct:
- Arithmetic Mean: The sum of deviations of items from the arithmetic mean ($\bar{x}$) always equals zero. That is, $\sum_{i=1}^{n} (x_i - \bar{x}) = 0$. However, the sum of the *absolute* deviations from the arithmetic mean, $\sum_{i=1}^{n} |x_i - \bar{x}|$, is generally larger than the sum of absolute deviations from the median. The arithmetic mean is known for minimizing the sum of *squared* deviations, $\sum_{i=1}^{n} (x_i - \bar{x})^2$.
- Harmonic Mean: The harmonic mean is typically used for averaging rates or ratios. It does not possess the property of minimizing the sum of absolute deviations.
- Mode: The mode is the most frequently occurring value in a dataset. Similar to the harmonic mean, the sum of absolute deviations from the mode is not generally minimized.
Conclusion
Based on the properties of measures of central tendency, the sum of the absolute deviations (i.e., deviations ignoring signs) of the items from the Median is the least.
Important Questions from Elementary Statistics (Notes)
Let $X_1, X_2, X_3$ be a random sample of size 3 from an absolutely continuous distribution that is symmetric about 0. For $i=1,2,3$, let $R_i$ denote the rank of $|X_i|$ among $|X_1|, |X_2|$ and $|X_3|$.
If $T^+ = \sum_{i=1, X_i>0}^3 R_i$
is the Willcoxon signed-rank statistic, then which of the following statements are true?,
- Which of the following is the first step in calculating the median of data set?
1. Average the middle two values of the data set
2. Array the data
3. Determine the relative weights of the data values in terms of importance
4. Find the average distance of the observations in the data set from the mean - The relationship between mean, median and mode is:
Which of the following statements are correct?
(A).When a cyclical pattern in data has a period of less than 1 year, the pattern in data is called seasonal variation
(B). When a cyclical pattern has a period more than 1 year, we refer to it as cyclical variation
(C). Seasonality is considered equivalent to forecasting
(D). Cyclical behaviour in business is also termed as business cycle
Choose the correct answer from the options given below:Match List-I with List-II
List-1 List-II Methods to Deseasonalise the Time Series Underlying Meaning (A). Method of simple average (I). It assumes that seasonal variation for a given month is constant fraction of trend (B). Ratio to Trend Method (II). It is the easiest method of obtaining a seasonal index (C). Ratio to moving average method (III). It is the most difficult method of measuring seasonal variations (D). Link relative method (IV). It is also known as the percentage of moving average method
Choose the correct answer from the options given below:
