GMAT Statistics & Average Part II

Welcome to the live session of Statistics & Average Part II. You can watch the entire session by playing the embedded video. The questions discussed on November 3, 2017 during the live session are listed below the embedded video. The video link next to each question will take you to the start of the part of the video where that specific question is discussed. Click the video embedded below to watch the entire session.

Concepts covered in this Free GMAT Live Session on Wizako's YouTube channel are

1. What is standard deviation? What does standard deviation measure?
2. How to compute standard deviation?
3. What does it mean when the standard deviation of a set of numbers is 0 (zero)?
4. What does it mean when we say observations lying within one standard deviation of the mean?
5. What is variance and how is it related to standard deviation?
6. Data Sufficiency questions (part of GMAT Data Insights section) in statistics and averages.

Watch this GMAT Standard Deviation and Data Sufficiency Part II | Youtube Live sessions

Play Video: GMAT Statistics and Averages Part II Live Session

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GMAT Questions Discussed in the Youtube Live Session

Standard Deviation

1. Consider the following set of numbers: 50, 40, 10, 20, 60, 30, 70, 90, 80. How many of these observations lie within one standard deviation of the mean?

   Correct Answer5 observations

   LOD: Easy

   Video Solution

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2. If the standard deviation of 4 positive numbers p, q, r, and s is d, what is the standard deviation of (2p − 2), (2q − 2), (2r − 2) and (2s − 2)?

   Correct AnswerThe standard deviation is 2d

   LOD: Easy

   Video Solution

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Data Sufficiency - Easy Questions

1. What is the median of integers a, b, and c?

   1. ab = 0
   2. bc = 0

   Correct AnswerChoice E: The two statements together are NOT sufficient.

   LOD: Easy Video Solution

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2. If ab = c, what is the median of a, b, and c?

   1. a = 1
   2. c = 1

   Correct AnswerChoice C: The two statements together are sufficient.

   LOD: Easy Video Solution

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3. If Set S = {x, x, y, z} what is the standard deviation of S?

   1. x = y = z
   2. x × y × z = 0

   Correct AnswerChoice A: Statement 1 ALONE is sufficient.

   LOD: Easy Video Solution

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4. A stationary car starts moving, and is accelerating constantly. Will it average higher than 60 km/h over a 4 km journey?

   1. The car travels the first 1 km in 1:30 minutes.
   2. The car travels the first 2 km in 2 minutes.

   Correct AnswerChoice B: Statement 2 ALONE is sufficient.

   LOD: Easy Video Solution

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5. What is the standard deviation of three positive integers, a, b, and c?

   1. a − b = b − c
   2. a = c

   Correct AnswerChoice C: Statements TOGETHER are sufficient.

   LOD: Easy Video Solution

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Data Sufficiency - Medium Difficulty Questions

1. a, b, and c are three positive integers such that abc = 12. What is the standard deviation of a, b, and c?

   1. The median of a, b, and c is equal to the range of a, b, and c.
   2. The median of a, b, and c is less than the range of a, b, and c.

   Correct AnswerChoice A: Statement 1 ALONE is sufficient.

   LOD: Medium Video Solution

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2. Is 'b' the median of three numbers, a, b, and c?

   1. |a − b| = |b − c|
   2. b > c

   Correct AnswerChoice E: Statements together are NOT sufficient.

   LOD: Medium Video Solution

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Method and Formula to Compute Standard Deviation

1. How to compute standard deviation for a set of numbers?

Let's learn the method by taking an example. What is the standard deviation of 1, 3, and 5?

Step 1: Compute the average of the given numbers.
Average of 1, 3, and 5 = \\frac{\text{1 + 3 + 5}}{3}) = \\frac{\text{9}}{3}) = 3

Step 2: Compute the Deviations for each of the terms. What is deviation?

Deviation is the difference between each of the terms and the average of the set of numbers.
The deviations for the 3 numbers are (1 − 3), (3 − 3), and (5 − 3).
The deviations are therefore -2, 0, and 2

Step 3: Square the Deviations
(-2)², 0², and (2)²
i.e., 4, 0, and 4

Step 4: Compute the Variance. What is variance?
The Average of the Squared Deviation is the Variance.
The average of 4, 0, and 4 = \\frac{\text{4 + 0 + 4}}{\text{3}}) = \\frac{\text{8}}{\text{3}})

Step 5: Compute Standard Deviation (SD). How to compute SD from Variance?
Standard Deviation is the square root of variance.
Hence, Standard Deviation of the given numbers is \\sqrt{\frac{\text{8}}{\text{3}}})

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2. What is the formula to compute the standard deviation of a set of numbers? The MOSSOM formula

Variance can be computed as the difference between the 'Mean of Squares' and the 'Square of Mean'.

Let's understand how to apply this formula by applying it to finding the standard deviation of x₁, x₂, and x₃

First part is mean of squares (MOS). \\frac{x_1^2 + x_2^2 + x_3^2}{3})

The second part is square of mean (SOM). \\left[ \frac{x_1 + x_2 + x_3}{3} \right]^2 )

Therefore, variance = \\frac{x_1^2 + x_2^2 + x_3^2}{3}) − \\left[ \frac{x_1 + x_2 + x_3}{3} \right]^2 )

And standard deviation = \\sqrt{\frac{x_1^2 + x_2^2 + x_3^2}{3} - \left[ \frac{x_1 + x_2 + x_3}{3} \right]^2 })

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3. When is the standard deviation equal to zero?

If all the numbers in the data set are equal, the standard deviation will be zero.