Solving Problems On Standard deviation

 

Understanding Standard Deviation

Standard Deviation is a fundamental statistical metric used to quantify the spread, dispersion, or variability within a dataset. While mean deviation deals with absolute distances, standard deviation uses squared differences from the mean. This mathematical property means outliers influence it heavily, making it an essential tool for statistical variance analysis.

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1. Ungrouped Data

Ungrouped data consists of raw, individual observations or measurements listed out in their original form (e.g., the test scores of a small group of students).

Formula (Sample Standard Deviation, s):

s = √[ Σ(x - x̄)² / (n - 1) ]

Formula (Population Standard Deviation, σ):

σ = √[ Σ(x - μ)² / N ]

Where:
• x = individual data values
• x̄ = sample mean / μ = population mean
• n = number of observations in the sample / N = total population size
• Σ(x - x̄)² = sum of the squared deviations from the mean

2. Grouped Data

Grouped data refers to information that has been organized into continuous frequency distributions or separated into distinct class intervals due to a large number of values.

Formula (Sample Standard Deviation, s):

s = √[ Σf(x - x̄)² / (Σf - 1) ]

Where:
• f = the frequency corresponding to each class/interval
• x = the midpoint of each specific class interval
• x̄ = the calculated grouped sample mean, represented by Σfx / Σf

Practice Questions (Ungrouped Data)

Test your understanding of sample standard deviation with these 5 Canadian-themed problems. Click "Show Answer & Solution" to check your work!

Q1. A youth hockey player in Ottawa registered the following point totals over 5 regular season games: 2, 4, 1, 3, and 5. Calculate the sample standard deviation of their points.

• A) 1.58 points
• B) 2.50 points
• C) 1.41 points
• D) 2.00 points

Correct Answer: A) 1.58 points

Solution:
1. Find the sample mean (x̄): (2 + 4 + 1 + 3 + 5) / 5 = 15 / 5 = 3.
2. Square the differences from this mean (3):
(2-3)² = 1; (4-3)² = 1; (1-3)² = 4; (3-3)² = 0; (5-3)² = 4.
3. Sum these squared differences: 1 + 1 + 4 + 0 + 4 = 10.
4. Divide by (n - 1), which is (5 - 1 = 4): 10 / 4 = 2.5.
5. Take the square root: √2.5 ≈ 1.58.

Q2. During a severe winter cold snap in Winnipeg, nighttime low temperatures across 4 consecutive days dropped to -12°C, -10°C, -6°C, and -4°C. Find the sample standard deviation of these temperatures.

• A) 3.16°C
• B) 3.83°C
• C) 2.95°C
• D) 4.10°C

Correct Answer: B) 3.83°C

Solution:
1. Determine the mean temperature (x̄): (-12 + -10 + -6 + -4) / 4 = -32 / 4 = -8°C.
2. Compute squared differences from the mean (-8):
(-12 - (-8))² = (-4)² = 16
(-10 - (-8))² = (-2)² = 4
(-6 - (-8))² = (2)² = 4
(-4 - (-8))² = (4)² = 16
3. Add the squared values together: 16 + 4 + 4 + 16 = 40.
4. Divide by (n - 1), which is (4 - 1 = 3): 40 / 3 = 13.333...
5. Take the square root: √13.333 ≈ 3.83°C.

Q3. A marine biologist tracking Orca populations off the coast of Vancouver Island recorded pod sizes of 6, 9, and 3 across three distinct sightings. Determine the sample standard deviation of these pod sizes.

• A) 2.5 pods
• B) 4.0 pods
• C) 3.0 pods
• D) 3.5 pods

Correct Answer: C) 3.0 pods

Solution:
1. Calculate the mean size (x̄): (6 + 9 + 3) / 3 = 18 / 3 = 6.
2. Obtain squared differences from the mean (6):
(6 - 6)² = 0; (9 - 6)² = 9; (3 - 6)² = 9.
3. Sum the calculated squares: 0 + 9 + 9 = 18.
4. Divide by (n - 1), which is (3 - 1 = 2): 18 / 2 = 9.
5. Find the final square root: √9 = 3.

Q4. The prices of a medium double-double coffee across four separate Tim Hortons storefronts in Nova Scotia are listed as $1.80, $1.90, $2.10, and $2.20. Find the sample standard deviation.

• A) $0.18
• B) $0.05
• C) $0.15
• D) $0.22

Correct Answer: A) $0.18

Solution:
1. Discover the average price (x̄): (1.80 + 1.90 + 2.10 + 2.20) / 4 = 8.00 / 4 = $2.00.
2. Evaluate the squared errors around the mean (2.00):
(1.80 - 2.00)² = 0.04; (1.90 - 2.00)² = 0.01; (2.10 - 2.00)² = 0.01; (2.20 - 2.00)² = 0.04.
3. Combine the squares: 0.04 + 0.01 + 0.01 + 0.04 = 0.10.
4. Divide by (n - 1), which is (4 - 1 = 3): 0.10 / 3 ≈ 0.03333.
5. Calculate the square root: √0.03333 ≈ $0.18.

Q5. Five post-secondary students in Montreal reported spending the following number of hours preparing for final exams over the weekend: 4, 7, 5, 9, and 5. What is the sample standard deviation of their study hours?

• A) 2.50 hours
• B) 1.73 hours
• C) 2.00 hours
• D) 1.95 hours

Correct Answer: C) 2.00 hours

Solution:
1. Find the average weekend study hours (x̄): (4 + 7 + 5 + 9 + 5) / 5 = 30 / 5 = 6 hours.
2. Figure out the squared variances from the mean (6):
(4 - 6)² = 4; (7 - 6)² = 1; (5 - 6)² = 1; (9 - 6)² = 9; (5 - 6)² = 1.
3. Tally the squared variations: 4 + 1 + 1 + 9 + 1 = 16.
4. Divide by (n - 1), which is (5 - 1 = 4): 16 / 4 = 4.
5. Finish with the square root calculation: √4 = 2.00 hours.