Variance and Standard Deviation Help (page 2)

By — McGraw-Hill Professional
Updated on Sep 13, 2011

Practice 2

What is the standard deviation of the distribution of scores for the quiz? Round off the answer to three decimal places.

Solution 2

The standard deviation, symbolized σ, is equal to the square root of the variance. We already know the variance, which is 4.204. The square root of this is found easily using a calculator. For best results, take the square root of the original quotient:

σ = (546:55=130)1/2

= 2:050

Practice 3

Draw a point-to-point graph of the distribution of quiz scores in the situation we've been dealing with in the past few paragraphs, showing the mean, and indicating the range of scores within 1 standard deviation of the mean (μ ± σ).

Solution 3

First, calculate the values of the mean minus the standard deviation (μ – σ) and the mean plus the standard deviation (μ + σ):

μ – σ = 5.623 – 2.050 = 3.573

μ + σ = 5.623 + 2.050 = 7.673

These are shown, along with the value of the mean itself, in Fig. 8-6. The scores that fall within this range are indicated by enlarged dots with light interiors. The scores that fall outside this range are indicated by smaller, solid black dots.

Variance and Standard Deviation

Fig. 8-6. Illustration for Practice 3.

Practice 4

In the quiz scenario we've been analyzing, how many students got scores within one standard deviation of the mean? What percentage of the total students does this represent?

Solution 4

The scores that fall within the range μ ± σ are 4, 5, 6, and 7. Referring to the original data portrayed in Table 8-1, we can see that 15 students got 4 answers right, 24 students got 5 answers right, 22 students got 6 answers right, and 24 students got 7 answers right.

Table 8-1 Table for Practice 4. The lowest score is at the tope and the highest score is at the bottom.

Therefore, the total number, call it nσ, of students who got answers within the range μ ± σ is simply the sum of these:

nσ = 15 + 24 + 22 + 24

= 85

The total number of students in the class, n, is equal to 130. To calculate the percentage of students, call it nσ%, who scored in the range μ ± σ, we must divide nσ by n and then multiply by 100:

nσ% = 100(nσ/n)

= 100(85/130)

= 65.38%

More practice problems for these concepts can be found at:

Statistics Practical Problems Practice Test

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