Sample Variance and Standard Deviation: Six Pulse Counts

1. Formula
   Observation

   The screen displays "Arithmetic mean is x̄ = (71+83+67+74+70+79)/6 = 74"

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

$\bar{x}$

Sample arithmetic mean

Real numbers

1. Formula
   Observation

   The screen displays "S² = 1/(6-1) [(71-74)² + ... + (79-74)²]"

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

S^2

Sample variance

Non-negative real numbers

1. Formula
   Observation

   The screen displays "The arithmetic mean is x̄ = (71+83+67+74+70+79)/6 = 74".

$\bar{x}$

Arithmetic mean of the sample data

Real numbers

1. Formula
   Observation

   The screen displays "S² = 1/(6-1)[(71-74)^2 + ... + (79-74)^2]", and finally calculates 36.

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

S^2

Sample variance

Non-negative real numbers

1. Formula
   Observation

   The screen displays "The sample standard deviation is S = √36 = 6".

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

S

Sample standard deviation

Non-negative real numbers

1. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

2. Formula
   Observation

   The denominator is written as 6-1, corresponding to sample size n=6.

n

Sample size

Positive integers

1. Formula
   Observation

   The screen provides the specific expansion of S² = 1/(n-1) Σ(x_i - x̄)², with the denominator explicitly written as 6-1

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

The source uses the corrected sample-variance definition: sum squared deviations from the sample mean and divide by n−1. This problem computes sample statistics; it does not determine the exact class population variance or establish an unbiased standard deviation.

1. Use the corrected sample-variance convention in this source

2. Sample size n is greater than 1

3. Use the mean of this same sample

1. Calculation of Arithmetic Mean

1. Formula
   Observation

   The screen displays x̄ = (71+83+67+74+70+79)/6 = 74

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

Sum all sample observations and divide by the sample size n to obtain the center of the sample. This step is a necessary prerequisite for calculating variance.

1. All sample observations must be known

1. Formula
   Observation

   The screen gives S² = 1/(6-1)[(71-74)^2+(83-74)^2+(67-74)^2+(74-74)^2+(70-74)^2+(79-74)^2].

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

The video demonstrates sample variance with specific data: first subtract the mean from each observation, then square the result, and finally sum these squared differences and divide by n-1. Here n=6, so the denominator is 6-1=5.

1. Sample data x_1,x_2,…,x_n must be known

2. The sample mean x̄ must be calculated first

3. The denominator uses n-1 instead of n

1. Arithmetic mean

1. Formula
   Observation

   The screen displays "The sample standard deviation is S = √36 = 6".

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

The video directly defines the sample standard deviation as the square root of the sample variance. Since S²=36 was calculated earlier, S=√36=6.

1. S² has been calculated first

2. Take the non-negative square root

1. Formula for calculating sample variance

1. Formula
   Observation

   The screen displays x̄=(71+83+67+74+70+79)/6=74.

The video first calculates the arithmetic mean of the six data points as the baseline for subsequent calculations of deviations from the mean.

1. Data consists of a finite number of values x_1,…,x_n

1. Formula
   Observation

   The screen step-by-step shows the summation of the numerator and division by 6, with the final result being 74

1. Expression

   $$\bar{x} = \frac{71+83+67+74+70+79}{6}$$
   Explanation

   Substitute the six pulse rate data points

   Justification

   Definition of arithmetic mean

   Shown in the video
2. Expression

   $$\bar{x} = 74$$
   Explanation

   Calculate the sum 444 divided by 6

   Justification

   Basic arithmetic operation

   Shown in the video

The sample mean is 74

1. Formula
   Observation

   The screen displays S² = 1/(6-1) [(71-74)² + (83-74)² + (67-74)² + (74-74)² + (70-74)² + (79-74)²]

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

1. The current 0–71-second interval sets up the formula; the full source later gives 36 and 6.

1. Expression

   $$S^2 = \frac{1}{6-1} \sum_{i=1}^{6} (x_i - 74)^2$$
   Explanation

   Substitute n=6 and x̄=74 into the sample variance formula

   Justification

   Definition of sample variance

   Shown in the video
2. Expression

   $$S^2 = \frac{1}{5} [(71-74)^2 + (83-74)^2 + (67-74)^2 + (74-74)^2 + (70-74)^2 + (79-74)^2]$$
   Explanation

   Expand the squared deviation of each observation from the mean

   Justification

   Expansion of summation notation

   Shown in the video

This sets up sample variance with denominator 5; the full source then completes the squared sum and obtains 36.

1. Formula
   Observation

   The screen writes the expansion and simplification of S² step by step, giving final value36.

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

1. Expression

   $$\bar{x}=\frac{71+83+67+74+70+79}{6}=74$$
   Explanation

   First calculate the mean of the six data points.

   Justification

   Definition of arithmetic mean.

   Shown in the video
2. Expression

   $$S^2=\frac{1}{6-1}[(71-74)^2+(83-74)^2+(67-74)^2+(74-74)^2+(70-74)^2+(79-74)^2]$$
   Explanation

   Subtract the mean from each data point, square the result, sum them up, and divide by 6-1.

   Justification

   Formula for sample variance.

   Shown in the video
3. Expression

   $$=\frac{1}{5}[(-3)^2+9^2+(-7)^2+0^2+(-4)^2+5^2]$$
   Explanation

   Calculate each deviation from the mean: 71-74=-3, 83-74=9, 67-74=-7, 74-74=0, 70-74=-4, 79-74=5.

   Justification

   Algebraic simplification.

   Shown in the video
4. Expression

   $$=\frac{1}{5}\times 180=36$$
   Explanation

   The sum of squares is 9+81+49+0+16+25=180, then multiply by 1/5.

   Justification

   Arithmetic calculation.

   Shown in the video

Sample variance S²=36.

1. Formula
   Observation

   The screen displays S=√36=6.

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

1. Expression

   $$S=\sqrt{S^2}$$
   Explanation

   Sample standard deviation is defined as the square root of sample variance.

   Justification

   Relationship between standard deviation and variance.

   Shown in the video
2. Expression

   $$S=\sqrt{36}=6$$
   Explanation

   Substitute S²=36 obtained in the previous step.

   Justification

   Arithmetic calculation.

   Shown in the video

Sample standard deviation S=6.

1. Caption evidence
   Observation

   Problem text: The commerce class has 40 students. A random sample of 6 students' pulse rates per minute are 71, 83, 67, 74, 70, 79.

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

1. The first two steps are retained from 0–71 seconds; the full numerical results appear later in the complete source.

A class has40 students. A random sample of6 students provides pulse data; find the sample variance and sample standard deviation.

1. Population size N=40 (used only to describe the sampling context, does not participate in sample statistic calculation)

2. Sample data x_i ∈ {71, 83, 67, 74, 70, 79}

3. Sample size n=6

Calculate sample variance S² and sample standard deviation S

1. Expression

   $$\bar{x} = \frac{71+83+67+74+70+79}{6} = 74$$
   Explanation

   First calculate the sample mean as the center for the variance formula

   Justification

   Definition of arithmetic mean

   Shown in the video
2. Expression

   $$S^2 = \frac{1}{6-1} [(71-74)^2 + (83-74)^2 + (67-74)^2 + (74-74)^2 + (70-74)^2 + (79-74)^2]$$
   Explanation

   Substitute into the sample variance formula, noting the denominator is n-1=5

   Justification

   Definition of sample variance

   Shown in the video

Full-source answer: sample variance 36 and sample standard deviation 6. This item retains the first two formula-setup steps.

Independent calculation gives squared-deviation sum 180, divided by 5 to get 36; its nonnegative square root is 6, matching the complete source result.

1. Caption evidence
   Observation

   Problem text: "The commerce class has 40 students. A random sample of 6 students' pulse rates per minute are 71, 83, 67, 74, 70, 79. Then the sample variance of the pulse rate is ______, and the sample standard deviation is ______."

2. Formula
   Observation

   The solution area step-by-step calculates the mean 74, variance 36, and standard deviation 6.

3. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

Given a sample of 6 students' pulse rates per minute as 71, 83, 67, 74, 70, 79, find the sample variance and sample standard deviation.

1. Sample data: 71, 83, 67, 74, 70, 79

2. Sample size n=6

3. Total population size 40 is background information only and does not participate in the calculation

Find S² and S.

1. Expression

   $$\bar{x}=\frac{71+83+67+74+70+79}{6}=74$$
   Explanation

   First calculate the mean.

   Justification

   Definition of arithmetic mean.

   Shown in the video
2. Expression

   $$S^2=\frac{1}{5}[(71-74)^2+(83-74)^2+(67-74)^2+(74-74)^2+(70-74)^2+(79-74)^2]$$
   Explanation

   Apply the formula for sample variance.

   Justification

   Definition of sample variance.

   Shown in the video
3. Expression

   $$S^2=\frac{1}{5}[(-3)^2+9^2+(-7)^2+0^2+(-4)^2+5^2]=\frac{1}{5}\times180=36$$
   Explanation

   Calculate the sum of squares and simplify.

   Justification

   Arithmetic operations.

   Shown in the video
4. Expression

   $$S=\sqrt{36}=6$$
   Explanation

   Take the square root of the variance.

   Justification

   Definition of standard deviation.

   Shown in the video

The sample variance is 36, and the sample standard deviation is 6.

The screen finally fills the answers back into the blanks in the problem: fill 36 in the variance blank and 6 in the standard deviation blank.

1. Animation
   Observation

   In the white writing area, the mean formula and variance expansion appear sequentially; key numbers 74 and denominator 6-1 are clearly visible

1. Mean formula

2. Variance expansion

3. Yellow circular 'Solution' label

1. From blank writing area to displaying complete mean calculation

2. From mean result to displaying variance formula structure

1. Problem statement text remains fixed at the top of the screen

2. Publisher watermark position unchanged

Visually presents the logical sequence from raw data to statistical formulas, emphasizing the structural feature of the n-1 denominator

1. Animation
   Observation

   The yellow cursor first stays on the denominator 6-1.

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

1. Denominator 6-1

2. Yellow cursor

1. Cursor stops at the denominator position, indicating that the denominator here is not 6 but 6-1

1. The numerator structure remains the sum of squared deviations from the mean

Visually highlights the difference in the denominator between sample variance and population variance: n-1 is used here.

1. Animation
   Observation

   The cursor moves to the mean 74, then to the first parenthesis (71-74)^2.

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

1. Mean 74

2. (71-74)^2

3. Yellow cursor

1. Cursor points to 74 first, then to 71-74

1. All parentheses adopt the form 'data minus mean'

Explains that each term in the variance is the difference between a single data point and the mean.

1. Animation
   Observation

   The cursor sequentially scans (83-74)^2, (67-74)^2, (74-74)^2, (70-74)^2, (79-74)^2.

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

1. Six squared difference terms

2. Yellow cursor

1. Cursor points to each term sequentially according to the order of the data in the problem

1. Each term is (x_i-74)^2

2. There are six terms in total, corresponding to n=6

Maps the abstract formula to concrete data, showing that variance sums the squared deviations of all samples.

1. Animation
   Observation

   A new line =1/5[(-3)^2+9^2+(-7)^2+0^2+(-4)^2+5^2] appears on the screen, and the cursor pauses on each term again.

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

1. Newly appeared simplified line

2. Values -3, 9, -7, 0, -4, 5

1. Transforms from the form (x_i-74)^2 to the specific squared difference form

1. Number of terms remains six

2. Denominator remains 5

Visually displays the result of algebraic simplification, making the subsequent sum of squares calculation clearer.

1. Animation
   Observation

   First "The sample standard deviation is S=√36=6" appears, then the blanks in the problem are filled with red text 36 and 6.

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

1. New standard deviation line at the bottom

2. Two blanks in the problem

3. Red answers 36, 6

1. Final standard deviation equation added to the solution area

2. Blanks in the problem change from empty to 36 and 6

1. Variance calculation result remains 36

2. Standard deviation calculation result remains 6

Maps the derived results back to the original problem, completing the fill-in-the-blank solution.

1. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

2. Formula
   Observation

   The variance denominator on the screen is written as 6-1 instead of 6

Believing that sample variance should also be divided by the sample size n

This problem explicitly uses corrected sample variance with denominator n−1. To describe a complete finite population, its variance instead divides squared deviations by the population size. This distinction does not assert that a standard-deviation estimator is unbiased or that the sample values determine the whole class population parameters.

1. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

2. Animation
   Observation

   The cursor stops on the denominator 6-1.

It is easy to mistakenly write the denominator of sample variance as the sample size n.

The video clearly states that this is sample variance, and the denominator should be n-1; in this problem n=6, so the denominator is 6-1=5.

1. Formula
   Observation

   The term (x_i - 74)² appears repeatedly in the variance expansion, where 74 is exactly the x̄ calculated in the previous step

The calculation of sample variance relies on the previously obtained sample mean as the baseline for deviations

1. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

Defines the denominator rule and scope of application for sample variance by contrasting it with population variance

1. Formula
   Observation

   First x̄=74 is given; then S² repeatedly uses terms subtracting74.

The calculation of sample variance depends on first finding the mean, and then subtracting that mean from each data point.

1. Formula
   Observation

   The screen transitions from S²=36 to S=√36=6.

2. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

The sample standard deviation is directly obtained by taking the square root of the sample variance.

1. Formula
   Observation

   The denominator of the S² formula is explicitly written as 6-1

1. Definition of Sample Variance
2. Confusion between Sample and Population Variance Denominators

1. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

1. Definition of Sample Variance
2. Confusion between Sample and Population Variance Denominators

1. Audio
   Observation

   The narration distinguishes sample and population, then calculates the pulse data through the sample mean, corrected sample variance and sample standard deviation.

1. Formula for calculating sample variance

1. Formula
   Observation

   Fully displays the calculation from the mean to the sum of squares and then to 36.

1. Arithmetic mean
2. Formula for calculating sample variance

1. Formula
   Observation

   S=√36=6.

1. Relationship between sample standard deviation and sample variance