Question
Consider the following for the next three (03) items that follow : The algebraic sum of the deviations of a set of values x 1, x 2, x 3, ⋯, x n measured from 100 is -20 and the algebraic sum of the deviations of the same set of values measured from 92 is 140.
What is the algebraic sum of the deviations of the same set of values measured from 99?
This question was previously asked in
NDA I 2022 GAT Previous Year Paper (10-Apr-2022)
The correct answer is
0
Understanding Algebraic Sum of Deviations
The question involves the concept of the algebraic sum of deviations. For a set of values \( x_1, x_2, \ldots, x_n \), the algebraic sum of deviations from a constant \( a \) is defined as \( \sum_{i=1}^{n} (x_i - a) \). This sum represents the total difference between each value in the set and the constant \( a \), taking into account the sign of the difference.Setting Up Equations from the Given Information
We are given information about the algebraic sum of deviations of a set of values from two different constants: 100 and 92. Let the set of values be \( x_1, x_2, \ldots, x_n \). The number of values in the set is \( n \). 1. The algebraic sum of deviations from 100 is -20. This can be written mathematically as: \( \sum_{i=1}^{n} (x_i - 100) = -20 \) 2. The algebraic sum of deviations from 92 is 140. This can be written mathematically as: \( \sum_{i=1}^{n} (x_i - 92) = 140 \) We need to find the algebraic sum of deviations of the same set of values measured from 99, which is \( \sum_{i=1}^{n} (x_i - 99) \). Let's expand the given summation equations. Remember that \( \sum_{i=1}^{n} (a+b) = \sum_{i=1}^{n} a + \sum_{i=1}^{n} b \) and \( \sum_{i=1}^{n} c = nc \) for a constant \( c \). Equation 1: \( \sum_{i=1}^{n} (x_i - 100) = \sum_{i=1}^{n} x_i - \sum_{i=1}^{n} 100 = \sum_{i=1}^{n} x_i - 100n \) So, we have: \( \sum_{i=1}^{n} x_i - 100n = -20 \) (Equation A) Equation 2: \( \sum_{i=1}^{n} (x_i - 92) = \sum_{i=1}^{n} x_i - \sum_{i=1}^{n} 92 = \sum_{i=1}^{n} x_i - 92n \) So, we have: \( \sum_{i=1}^{n} x_i - 92n = 140 \) (Equation B) We now have a system of two linear equations with two unknowns, \( \sum x_i \) (the sum of the values) and \( n \) (the number of values).Solving for the Number of Values and Sum of Values
We can solve this system of equations to find the values of \( \sum x_i \) and \( n \). Let's subtract Equation B from Equation A: \( (\sum_{i=1}^{n} x_i - 100n) - (\sum_{i=1}^{n} x_i - 92n) = -20 - 140 \) \( \sum_{i=1}^{n} x_i - 100n - \sum_{i=1}^{n} x_i + 92n = -160 \) \( -8n = -160 \) Solving for \( n \): \( n = \frac{-160}{-8} \) \( n = 20 \) So, there are 20 values in the set. Now, substitute the value of \( n=20 \) back into either Equation A or Equation B to find \( \sum x_i \). Using Equation A: \( \sum_{i=1}^{n} x_i - 100(20) = -20 \) \( \sum_{i=1}^{n} x_i - 2000 = -20 \) \( \sum_{i=1}^{n} x_i = 2000 - 20 \) \( \sum_{i=1}^{n} x_i = 1980 \) The sum of the values in the set is 1980. Let's verify using Equation B: \( \sum_{i=1}^{n} x_i - 92(20) = 140 \) \( \sum_{i=1}^{n} x_i - 1840 = 140 \) \( \sum_{i=1}^{n} x_i = 1840 + 140 \) \( \sum_{i=1}^{n} x_i = 1980 \) The results are consistent.Calculating the Required Algebraic Sum from 99
We need to find the algebraic sum of the deviations of the same set of values measured from 99. This is \( \sum_{i=1}^{n} (x_i - 99) \). Expanding this sum: \( \sum_{i=1}^{n} (x_i - 99) = \sum_{i=1}^{n} x_i - \sum_{i=1}^{n} 99 \) \( \sum_{i=1}^{n} (x_i - 99) = \sum_{i=1}^{n} x_i - 99n \) We have found that \( \sum_{i=1}^{n} x_i = 1980 \) and \( n=20 \). Substitute these values into the expression: \( \sum_{i=1}^{n} (x_i - 99) = 1980 - 99(20) \) \( \sum_{i=1}^{n} (x_i - 99) = 1980 - 1980 \) \( \sum_{i=1}^{n} (x_i - 99) = 0 \) The algebraic sum of the deviations of the set of values measured from 99 is 0.Alternative Method Using the Mean Property
The mean of a set of values \( \bar{x} \) is defined as \( \bar{x} = \frac{\sum x_i}{n} \). The algebraic sum of deviations from any constant \( a \) can also be expressed in terms of the mean: \( \sum_{i=1}^{n} (x_i - a) = \sum_{i=1}^{n} (x_i - \bar{x} + \bar{x} - a) \) \( \sum_{i=1}^{n} (x_i - a) = \sum_{i=1}^{n} (x_i - \bar{x}) + \sum_{i=1}^{n} (\bar{x} - a) \) The sum of deviations from the mean, \( \sum_{i=1}^{n} (x_i - \bar{x}) \), is always 0. The second term is the sum of a constant \( (\bar{x} - a) \) repeated \( n \) times: \( \sum_{i=1}^{n} (\bar{x} - a) = n(\bar{x} - a) \). So, the formula is: \( \sum_{i=1}^{n} (x_i - a) = n(\bar{x} - a) \). Using this property, the given information translates to: 1. \( \sum_{i=1}^{n} (x_i - 100) = n(\bar{x} - 100) = -20 \) 2. \( \sum_{i=1}^{n} (x_i - 92) = n(\bar{x} - 92) = 140 \) We can divide the first equation by the second to eliminate \( n \) and solve for \( \bar{x} \): \( \frac{n(\bar{x} - 100)}{n(\bar{x} - 92)} = \frac{-20}{140} \) \( \frac{\bar{x} - 100}{\bar{x} - 92} = \frac{-1}{7} \) Cross-multiply: \( 7(\bar{x} - 100) = -1(\bar{x} - 92) \) \( 7\bar{x} - 700 = -\bar{x} + 92 \) \( 7\bar{x} + \bar{x} = 92 + 700 \) \( 8\bar{x} = 792 \) \( \bar{x} = \frac{792}{8} \) \( \bar{x} = 99 \) The mean of the set of values is 99. Now we need to find the algebraic sum of deviations from 99, which is \( \sum_{i=1}^{n} (x_i - 99) \). Using the property \( \sum_{i=1}^{n} (x_i - a) = n(\bar{x} - a) \): \( \sum_{i=1}^{n} (x_i - 99) = n(\bar{x} - 99) \) Since we found \( \bar{x} = 99 \): \( \sum_{i=1}^{n} (x_i - 99) = n(99 - 99) \) \( \sum_{i=1}^{n} (x_i - 99) = n(0) \) \( \sum_{i=1}^{n} (x_i - 99) = 0 \) Both methods show that the algebraic sum of the deviations of the set of values measured from 99 is 0. This is consistent with the property that the algebraic sum of deviations from the mean is always zero. The final answer is 0.| Step | Description | Calculation/Formula | Result |
|---|---|---|---|
| 1 | Set up equations from given information | \( \sum (x_i - 100) = -20 \) \( \sum (x_i - 92) = 140 \) |
\( \sum x_i - 100n = -20 \) \( \sum x_i - 92n = 140 \) |
| 2 | Solve for number of values (n) | Subtract eqns: \( -8n = -160 \) | \( n = 20 \) |
| 3 | Solve for sum of values (\( \sum x_i \)) | Using \( n=20 \) in \( \sum x_i - 100n = -20 \) | \( \sum x_i = 1980 \) |
| 4 | Calculate sum of deviations from 99 | \( \sum (x_i - 99) = \sum x_i - 99n \) | \( 1980 - 99(20) = 0 \) |
Revision Table: Key Concepts in Deviations
| Concept | Definition | Formula | Property |
|---|---|---|---|
| Deviation | The difference between a value and a constant (or mean). | \( x_i - a \) | |
| Algebraic Sum of Deviations from a constant \( a \) | The sum of differences of each value from a constant \( a \). | \( \sum_{i=1}^{n} (x_i - a) \) | \( \sum (x_i - a) = \sum x_i - na \) |
| Algebraic Sum of Deviations from the Mean (\( \bar{x} \)) | The sum of differences of each value from the mean. | \( \sum_{i=1}^{n} (x_i - \bar{x}) \) | Always equals 0. |
| Mean (\( \bar{x} \)) | The average of the set of values. | \( \bar{x} = \frac{\sum x_i}{n} \) |
Additional Information: Properties of Deviations
Understanding deviations is fundamental in statistics, particularly in calculating measures of dispersion like variance and standard deviation. Here are some key properties:- The algebraic sum of deviations of a set of values from their arithmetic mean is always zero. This is a very important property used in many statistical proofs and calculations. As we saw in this problem, the point 99 turned out to be the mean, leading to the sum of deviations from 99 being zero.
- If the algebraic sum of deviations from a point \( a \) is zero, then \( a \) must be the arithmetic mean of the set of values.
- The sum of the squared deviations from the mean is the minimum possible sum of squared deviations from any point. This property is why the mean is used as the central point for calculating variance.
- The sum of deviations changes linearly with the reference point \( a \). The difference in the sum of deviations when shifting the reference point from \( a_1 \) to \( a_2 \) is \( \sum (x_i - a_2) - \sum (x_i - a_1) = (\sum x_i - na_2) - (\sum x_i - na_1) = na_1 - na_2 = n(a_1 - a_2) \). In this problem, the shift from 100 to 92 (a difference of 8) resulted in a change in sum from -20 to 140 (a difference of 160). \( 160 / 8 = 20 \), which is \( n \). The shift from 100 to 99 (a difference of 1) should result in a change of \( n(100-99) = 20(1) = 20 \). The sum from 100 was -20, so the sum from 99 should be -20 + 20 = 0. This confirms our result.
Was this answer helpful?
Similar Questions
The mean and variance of five observations are 14 and 13.2 respectively. Three of the five observations are 11, 16 and 20. What are the other two observations ?
A die is thrown 10 times and obtained the following outputs :
1, 2, 1, 1, 2, 1, 4, 6, 5, 4
What will be the mode of data so obtained ?
For data -1, 1, 4, 3, 8, 12, 17, 19, 9, 11; if M is the median of first 5 observations and N is the median of last five observations, then what is the value of 4M - N ?
Let P, Q, R represent mean, median and mode. If for some distribution \(5 P=4 Q=\frac{R}{2}\) then what is \(\frac{P+Q}{2 P+0.7 R}\) equal to ?
Consider the following frequency distribution :
x 1 2 3 5 f 4 6 9 7 What is the value of median of the distribution ?
What is the mean of the values?
Important Questions from Statistics
Match the following:
(a) Marginalist Revolution (i) Samuelson (b) Multiplier-Accelerator model (ii) J. R. Hicks (c) IS-LM curves (iii) Jevous (d) Real Business Cycle (iv) Robert J. Borro Choose the correct option from those given below:
Which one of the following responses is true as a solution to simultaneous equation bias?
A. OLS method
B. Principle Component Method
C. Two - stage Least Square Method (2 SLS method)
D. Full Information Maximum Likelihood method (FIML)
Choose the correct option.
Time series under the condition (E xt ) = μ and cov(x t, x t + k ) = Y(K) is said to be
Given the sample size 400 with the sample mean 99, the population mean 100 and computed value of z statistic at 2.5, the value of population standard deviation will be
Which one of the following price index numbers satisfies the factor reversal test?
Need Expert Advice?
Test Series
Defence
NDA 2026 Mock Test Series (Latest Pattern)
501 Tests 1 Tests Free
658 Attempts
4.7(120)
English, Hindi
More Questions from NDA
