Population standard deviation: a worked example

1. Formula
   Observation

   The second line of the screen displays "Thus σ₁² = ((6-8)² + (7-8)² + (8-8)² + (9-8)² + (10-8)²)/5 = 10/5 = 2".

2. Audio
   Observation

   The variance (square of the standard deviation) of this population dataset

\sigma_1^2

The variance (square of the standard deviation) of this population dataset

Non-negative real number; calculated as 2 in this problem

1. Formula
   Observation

   The third line of the screen displays "Therefore, the standard deviation is σ₁ = √2 ≈ 1.41".

2. Audio
   Observation

   The standard deviation of this population dataset

\sigma_1

The standard deviation of this population dataset

Non-negative real number; calculated as √2, approximately 1.41, in this problem

1. Formula
   Observation

   The first line of the screen displays "The mean of 6, 7, 8, 9, 10 is (6+7+8+9+10)/5 = 8".

2. Audio
   Observation

   The mean of the data 6, 7, 8, 9, 10

8

The mean of the data 6, 7, 8, 9, 10

Real number

1. Audio
   Observation

   The sequence demonstrated in the video is: first find the mean of the data, then use the squared differences between each data point and the mean, take their average to get the variance σ₁², and finally take the square root of σ₁² to get the standard deviation σ₁. The population version is used here, with the denominator being the number of data points, 5.

2. Formula
   Observation

   The screen presents three lines in order: the mean, the variance σ₁², and the standard deviation σ₁.

The sequence demonstrated in the video is: first find the mean of the data, then use the squared differences between each data point and the mean, take their average to get the variance σ₁², and finally take the square root of σ₁² to get the standard deviation σ₁. The population version is used here, with the denominator being the number of data points, 5.

1. The data is population data

2. Number of data points n=5

3. Find the mean first, then the sum of squared deviations from the mean

4. The descriptive population calculation applies to a nonempty finite collection of real-valued data; take the nonnegative square root.

1. 8
2. \sigma_1^2
3. \sigma_1

1. Formula
   Observation

   The screen clearly writes σ₁² = ((6-8)²+(7-8)²+(8-8)²+(9-8)²+(10-8)²)/5 = 10/5 = 2.

2. Audio
   Observation

   The video handles the standard deviation via its square σ₁² first. The formula content is the sum of squares of each data point minus the mean 8, divided by the number of data points 5. This corresponds to the definition of population variance.

The video handles the standard deviation via its square σ₁² first. The formula content is the sum of squares of each data point minus the mean 8, divided by the number of data points 5. This corresponds to the definition of population variance.

1. The mean has been previously calculated as 8

2. The denominator uses 5, not 4

1. 8

1. Formula
   Observation

   The third line of the screen displays "Therefore, the standard deviation is σ₁ = √2 ≈ 1.41".

2. Audio
   Observation

   The final step in the video takes the square root of the already calculated variance 2, obtaining the standard deviation √2, and provides the decimal approximation 1.41.

The final step in the video takes the square root of the already calculated variance 2, obtaining the standard deviation √2, and provides the decimal approximation 1.41.

1. σ₁² has been calculated first

2. Standard deviation takes the non-negative square root

1. \sigma_1^2
2. \sigma_1

1. Formula
   Observation

   The screen sequentially displays three lines of derivation: Mean=8; σ₁²=...=2; σ₁=√2≈1.41.

2. Audio
   Observation

   The standard deviation of this population dataset 6, 7, 8, 9, 10 is √2, approximately equal to 1.41.

1. Expression

   $$\frac{6+7+8+9+10}{5}=8$$
   Explanation

   First, calculate the mean of the five data points.

   Justification

   The first line of the screen and the narrator provide this formula simultaneously.

   Shown in the video
2. Expression

   $$\sigma_1^2=\frac{(6-8)^2+(7-8)^2+(8-8)^2+(9-8)^2+(10-8)^2}{5}$$
   Explanation

   Subtract the mean 8 from each data point, square them, sum them up, and divide by 5.

   Justification

   The left side of the second line on the screen completely lists this expression.

   Shown in the video
3. Expression

   $$=\frac{10}{5}=2$$
   Explanation

   The sum of the squared terms in the numerator simplifies to 10, and dividing by 5 yields the variance 2.

   Justification

   The right side of the second line on the screen directly writes the simplified result.

   Shown in the video
4. Expression

   $$\sigma_1=\sqrt{2}\approx 1.41$$
   Explanation

   Take the square root of the variance to get the standard deviation and provide a decimal approximation.

   Justification

   The third line on the screen and the narrator both provide this conclusion.

   Shown in the video

The standard deviation of this population dataset 6, 7, 8, 9, 10 is √2, approximately equal to 1.41.

1. Caption evidence
   Observation

   The problem area on the screen writes: "Let a set of population data have 5 numbers as follows: 6, 7, 8, 9, 10. Find the standard deviation of this dataset."

2. Audio
   Observation

   The lesson presents the specified population and shows the mean, variance and square-root result in a three-line solution.

3. Formula
   Observation

   After 00:14, the screen displays the complete three-line solution.

Let a set of population data have 5 numbers as follows: 6, 7, 8, 9, 10. Find the standard deviation of this dataset.

1. The data is population data

2. Number of data points is 5

3. Data values are 6, 7, 8, 9, 10

Find the standard deviation of this dataset.

1. Expression

   $$\frac{6+7+8+9+10}{5}=8$$
   Explanation

   First, calculate the mean.

   Justification

   First line of the video's solution.

   Shown in the video
2. Expression

   $$\sigma_1^2=\frac{(6-8)^2+(7-8)^2+(8-8)^2+(9-8)^2+(10-8)^2}{5}=\frac{10}{5}=2$$
   Explanation

   Calculate the average of the sum of squared deviations according to the population variance formula.

   Justification

   Second line of the video's solution.

   Shown in the video
3. Expression

   $$\sigma_1=\sqrt{2}\approx 1.41$$
   Explanation

   Take the square root of the variance to get the standard deviation.

   Justification

   Third line of the video's solution.

   Shown in the video

σ₁ = √2 ≈ 1.41

The video does not perform separate verification; it only derives the answer directly through three consecutive calculation lines.

1. Diagram
   Observation

   At the top of the screen is a green title "In-class Exercise". Below, the problem text is fully displayed. On the left is an icon with the character "Solution", but the solution area is blank.

2. Animation
   Observation

   The cursor moves near the problem statement; no formulas have appeared yet.

1. Title "In-class Exercise"

2. Problem text

3. "Solution" icon

4. Bottom chapter label "3-1 Univariate Data Analysis"

1. Cursor moves over the problem stem

2. Solution area remains blank until approximately 00:14

1. Problem stem text unchanged

2. Page layout structure unchanged

Visually, the problem is presented completely first before entering the calculation, helping learners recognize that this is an exercise asking for the standard deviation of population data.

1. Animation
   Observation

   At approximately 00:14, three lines of formulas appear at once in the solution area.

2. Formula
   Observation

   The three lines are respectively the final results for the mean, σ₁², and σ₁.

1. Mean formula

2. Variance formula

3. Standard deviation formula

4. Cursor

1. Blank solution area replaced by three complete formula lines

2. Cursor subsequently moves between the three lines to indicate

1. Problem stem remains at the top

2. Once the three formula lines appear, they are not rewritten

The screen compresses the solution process into three key steps: first mean, then variance, finally square root, emphasizing the fixed order of standard deviation calculation.

1. Caption evidence
   Observation

   The problem stem explicitly writes "population data".

2. Formula
   Observation

   The denominator of the variance in the solution uses 5.

Learners might confuse population standard deviation with sample standard deviation, mistakenly thinking the denominators are the same.

This problem gives a full population, so its variance uses n=5. When estimating population variance from a sample, the common Bessel-corrected sample variance instead uses n-1. That inferential setting is not this exercise.

1. Formula
   Observation

   Each term in the second line is (data value - 8)², where 8 comes from the mean result in the first line.

2. Audio
   Observation

   The variance calculation directly depends on the previously calculated mean 8; without the mean, the sum of squared deviations cannot be written.

The variance calculation directly depends on the previously calculated mean 8; without the mean, the sum of squared deviations cannot be written.

1. Formula
   Observation

   The second line first obtains σ₁²=2, and the third line writes σ₁=√2≈1.41.

2. Audio
   Observation

   Standard deviation is obtained by taking the square root of the variance as a subsequent application step.

Standard deviation is obtained by taking the square root of the variance as a subsequent application step.

1. Formula
   Observation

   The entire page solution applies the three-step method for population standard deviation to the data in the problem stem.

This example specifically demonstrates the use of the population standard deviation calculation procedure on actual data.

1. Audio
   Observation

   The displayed calculation answers the question: How to calculate the standard deviation of a set of population data?

2. Formula
   Observation

   The three lines on the screen correspond to these three steps in order.

1. Calculation procedure for population standard deviation
2. Variance formula used in this problem
3. Obtaining standard deviation by taking the square root of variance

1. Caption evidence
   Observation

   The problem stem states this is "population data".

2. Formula
   Observation

   The denominator in the variance formula is 5.

1. Variance formula used in this problem
2. Different denominators for population and sample

1. Formula
   Observation

   The final answer is written on the third line: σ₁=√2≈1.41.

1. In-class exercise: Find the standard deviation of population data 6, 7, 8, 9, 10
2. Obtaining standard deviation by taking the square root of variance