Population Variance and Standard Deviation: Five Weights

1. Formula
   Observation

   The screen displays the arithmetic mean as μ = (67+79+55+61+88)/5 = 70

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

μ

Arithmetic mean (in this problem, the average weight of the five team members)

Real numbers

1. Formula
   Observation

   The screen displays the population variance as σ² = 1/5[(67-70)² + (79-70)² + (55-70)² + (61-70)² + (88-70)²]

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

σ²

Population variance

Non-negative real numbers

1. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

2. Formula
   Observation

   The denominator in the formula is written as 5

n

Number of data points in the population (number of team members)

Positive integers

1. Formula
   Observation

   The screen displays "The arithmetic mean is μ = (67+79+55+61+88)/5 = 70".

$\mu$

The arithmetic mean of the dataset, i.e., the population mean

Real numbers; in this problem, the average weight of the data, unit is kilograms

1. Formula
   Observation

   The screen displays "The population variance is σ² = 1/5[(67-70)² + (79-70)² + (55-70)² + (61-70)² + (88-70)²]", subsequently simplified to 144.

$\sigma^2$

Population variance, equal to the sum of squared deviations of each data point from the mean divided by the number of data points

Non-negative real numbers; the result for this problem is 144

1. Formula
   Observation

   The screen displays "The population standard deviation is σ = √144 = 12".

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

$\sigma$

Population standard deviation, equal to the non-negative square root of the population variance

Non-negative real numbers; the result for this problem is 12

1. Formula
   Observation

   The denominator in the mean and the coefficient in the variance both use 5.

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

n

Number of data points / Population size

Positive integer; in this problem n=5

1. Formula
   Observation

   The formula substitutes 67, 79, 55, 61, 88 one by one, subtracts 70, and squares the result.

x_i

The i-th data value; in this problem, the weights of the five members

Real numbers; values in this problem are 67, 79, 55, 61, 88

1. Formula
   Observation

   μ = (67+79+55+61+88)/5 = 70

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

The arithmetic mean μ is equal to the sum of all data values divided by the number of data points n. In this problem, the weights of the five team members are summed and divided by 5, resulting in an average weight of 70 kg.

1. Data consists of a finite number of numerical values

2. n > 0

1. Formula
   Observation

   σ² = 1/5[(67-70)² + (79-70)² + (55-70)² + (61-70)² + (88-70)²]

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

The population variance σ² is defined as the sum of the squares of the differences between each data value and the population mean μ, divided by the number of data points n in the population. In this problem, μ=70 is found first, then the squared differences between each weight and 70 are calculated and averaged.

1. The population mean μ is known

2. The data represents the entire population rather than a sample

1. Arithmetic Mean

1. Formula
   Observation

   The screen introduces σ² = 1/5[(67-70)² + (79-70)² + (55-70)² + (61-70)² + (88-70)²] with the label "Population variance is".

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

The video writes the population variance as the average of the squared deviations of the data from the mean: first subtract the mean μ from each data point x_i, then square it, and finally divide by the number of data points n. In this problem, n=5 and μ=70.

1. The data is treated as the entire population rather than a sample

2. μ is the mean of that population

3. n is the number of data points in the population

1. Arithmetic Mean

1. Formula
   Observation

   The screen displays "The population standard deviation is σ = √144 = 12".

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

The video defines the population standard deviation as the square root of the population variance. Since the variance was calculated as 144, the standard deviation is √144=12.

1. σ² is the population variance

2. Take the non-negative square root

1. Definition of Population Variance

1. Formula
   Observation

   The screen displays μ=(67+79+55+61+88)/5=70.

The video first calculates the arithmetic mean of the five data points to serve as the baseline for subsequent deviation calculations.

1. n is the number of data points

2. In this problem n=5

1. Formula
   Observation

   Step-by-step display shows the numerator as 67+79+55+61+88, the denominator as 5, and the result as 70

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

1. Expression

   $$\mu = \frac{67+79+55+61+88}{5}$$
   Explanation

   Substitute the weights of the five team members into the arithmetic mean formula

   Justification

   Definition of arithmetic mean

   Shown in the video
2. Expression

   $$\mu = \frac{350}{5} = 70$$
   Explanation

   Calculate the sum of the numerator and divide by 5

   Justification

   Basic arithmetic operations

   Derived from the video

The average weight of the five team members is 70 kg.

1. Formula
   Observation

   The screen fully expands σ² = 1/5[(67-70)² + (79-70)² + (55-70)² + (61-70)² + (88-70)²]

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

1. The0–60second interval sets up the formula; the full source later completes the results144 and12.

1. Expression

   $$\sigma^2 = \frac{1}{5}[(67-70)^2 + (79-70)^2 + (55-70)^2 + (61-70)^2 + (88-70)^2]$$
   Explanation

   Substitute each weight and the previously calculated mean of 70 into the population variance formula

   Justification

   Definition of population variance

   Shown in the video

This sets up the population variance; the subsequent calculation of squared deviations gives the full-source result144 square kilograms.

1. Formula
   Observation

   The screen step-by-step shows σ² = 1/5[(67-70)²+(79-70)²+(55-70)²+(61-70)²+(88-70)²] = 1/5[(-3)²+9²+(-15)²+(-9)²+18²] = 1/5×720 = 144.

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

1. Expression

   $$\mu=\frac{67+79+55+61+88}{5}=70$$
   Explanation

   First calculate the mean to serve as the baseline for each data point's deviation.

   Justification

   Definition of arithmetic mean.

   Shown in the video
2. Expression

   $$\sigma^2=\frac{1}{5}\left[(67-70)^2+(79-70)^2+(55-70)^2+(61-70)^2+(88-70)^2\right]$$
   Explanation

   Subtract the mean 70 from each of the five data points, square the results, add them up, and finally divide by 5.

   Justification

   Definition of population variance.

   Shown in the video
3. Expression

   $$=\frac{1}{5}\left[(-3)^2+9^2+(-15)^2+(-9)^2+18^2\right]$$
   Explanation

   Calculate the differences inside the parentheses: 67-70=-3, 79-70=9, 55-70=-15, 61-70=-9, 88-70=18.

   Justification

   Algebraic simplification.

   Shown in the video
4. Expression

   $$=\frac{1}{5}\times 720$$
   Explanation

   Sum the five squared terms to get 720.

   Justification

   Arithmetic summation; the video does not explicitly show the intermediate addition of 9, 81, 225, 81, 324, but the result can be verified.

   Shown in the video
5. Expression

   $$=144$$
   Explanation

   Complete the division to obtain the population variance.

   Justification

   720÷5=144.

   Shown in the video

The population variance for this problem is 144.

1. Formula
   Observation

   The screen displays σ=√144=12.

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

1. Expression

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

   Standard deviation is defined as the square root of the variance.

   Justification

   Definition of population standard deviation.

   Shown in the video
2. Expression

   $$=\sqrt{144}$$
   Explanation

   Substitute the variance 144 calculated in the previous step.

   Justification

   Substitution of equal quantities.

   Shown in the video
3. Expression

   $$=12$$
   Explanation

   Calculate the square root.

   Justification

   12^2=144, and standard deviation takes the non-negative value.

   Shown in the video

The population standard deviation for this problem is 12.

1. Caption evidence
   Observation

   Problem text: A cheerleading squad has 5 members whose weights (unit: kg) are 67, 79, 55, 61, and 88 respectively. The population variance of the weights is ______, and the population standard deviation is ______.

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

1. These first two steps belong to0–60seconds; the complete answer is shown later in the full source.

Given the weights of 5 team members as 67, 79, 55, 61, and 88 kg, find the population variance and population standard deviation.

1. Dataset: {67, 79, 55, 61, 88}

2. Unit: kg

3. n = 5

Calculate population variance σ² and population standard deviation σ

1. Expression

   $$\mu = \frac{67+79+55+61+88}{5} = 70$$
   Explanation

   First find the arithmetic mean as the baseline for variance calculation

   Justification

   Variance definition depends on the mean

   Shown in the video
2. Expression

   $$\sigma^2 = \frac{1}{5}[(67-70)^2 + (79-70)^2 + (55-70)^2 + (61-70)^2 + (88-70)^2]$$
   Explanation

   Substitute into the population variance formula and expand the squared deviations from the mean

   Justification

   Definition of population variance

   Shown in the video

Full-source answer: population variance144 square kilograms and population standard deviation12 kilograms. This item retains the first two setup steps.

The correctness of the formula expansion can be verified by checking whether each squared deviation correctly corresponds to the original data and the mean of 70.

1. Formula
   Observation

   The problem text gives the weights of five members as 67, 79, 55, 61, 88, asking for the population variance and population standard deviation.

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

A cheerleading squad has 5 members with weights (unit: kg) of 67, 79, 55, 61, and 88 respectively. Find the population variance and population standard deviation of the weights.

1. Data represents the entire population

2. n=5

3. x_1=67, x_2=79, x_3=55, x_4=61, x_5=88

Find σ² and σ.

1. Expression

   $$\mu=\frac{67+79+55+61+88}{5}=70$$
   Explanation

   First calculate the mean.

   Justification

   Definition of arithmetic mean.

   Shown in the video
2. Expression

   $$\sigma^2=\frac{1}{5}\left[(67-70)^2+(79-70)^2+(55-70)^2+(61-70)^2+(88-70)^2\right]$$
   Explanation

   Apply the population variance formula.

   Justification

   Definition of population variance.

   Shown in the video
3. Expression

   $$=\frac{1}{5}\left[(-3)^2+9^2+(-15)^2+(-9)^2+18^2\right]=\frac{1}{5}\times720=144$$
   Explanation

   Simplify differences, square, sum, and divide by 5.

   Justification

   Algebraic operations and arithmetic summation.

   Shown in the video
4. Expression

   $$\sigma=\sqrt{144}=12$$
   Explanation

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

   Justification

   Definition of population standard deviation.

   Shown in the video

The population variance is 144, and the population standard deviation is 12.

Reverse check: 12^2=144, and 144×5=720 matches the sum of the five squared deviations.

1. Diagram
   Observation

   A white slide with a yellow frame displays the complete problem text and two blank lines for filling in answers

2. Animation
   Observation

   A yellow cursor sequentially points to the numbers and keywords in the problem

1. Problem text

2. Blank lines

3. Yellow cursor

1. Cursor moves from the title to each weight value

2. Cursor points to the blanks for 'population variance' and 'population standard deviation'

1. Problem text content remains unchanged

2. Background layout remains unchanged

Visual guidance via the cursor focuses the learner on key data and the solution goal.

1. Animation
   Observation

   After the circular icon with the character 'Solution' appears, the arithmetic mean formula emerges term by term

2. Animation
   Observation

   After the mean result 70 is displayed, the population variance formula is presented in its fully expanded form

1. 'Solution' icon

2. Mean formula

3. Expanded variance formula

1. Formula goes from blank to fully displayed

2. Value 70 is filled into the variance formula

1. Formula structure and symbols remain consistent

The animation synchronizes with the instructor's speech rate to gradually reveal the calculation logic, reinforcing the derivation order from mean to variance.

1. Animation
   Observation

   A yellow cursor sequentially points to (67-70)², (79-70)², (55-70)², (61-70)², (88-70)² in the σ² formula.

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

1. σ² formula

2. Five (x_i-70)² terms

3. Yellow cursor

1. Cursor moves from the first term to the fifth term

2. Each term is emphasized sequentially

1. Mean 70 remains unchanged

2. Number of data points 5 remains unchanged

3. Formula structure is sum of squared deviations divided by 5

Visually emphasizes that population variance is the average of the squared deviations of each data point relative to the mean.

1. Animation
   Observation

   The screen adds a line =1/5[(-3)²+9²+(-15)²+(-9)²+18²], then displays =1/5×720=144.

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

1. Original squared deviation expression

2. Simplified squared terms

3. 720

4. 144

1. Differences inside parentheses are calculated

2. Sum of squares is combined into 720

3. Dividing by 5 yields 144

1. Still the same σ² expression

2. Denominator remains 5

The animation transforms the abstract formula into concrete numerical calculation, explaining that the final value of variance comes from the average of the total squared deviations.

1. Animation
   Observation

   The blanks in the problem are first filled with 144, then a new line "Population standard deviation is σ=√144=12" appears below, and finally the second blank is filled with 12.

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

1. Two blanks in the problem

2. σ=√144=12 formula

3. Numbers 144 and 12

1. First blank changes from empty to 144

2. New standard deviation calculation line added

3. Second blank changes from empty to 12

1. The data itself remains unchanged

2. Variance result 144 remains unchanged

The visual sequence corresponds to the problem-solving flow: first find the variance, then derive the standard deviation from the variance, and finally fill the answers back into the problem blanks.

1. Formula
   Observation

   The problem explicitly states "population variance", and the formula on screen uses 1/5 instead of 1/(5-1).

1. The sample-population denominator contrast is editorial; the source explicitly uses the population formula.

Dividing all variances by n-1.

This problem involves population data, so the video adopts σ²=(1/n)Σ(x_i-μ)², meaning the denominator is n=5; if it were sample variance, the formula and denominator would be different.

1. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

2. Formula
   Observation

   The 70 in the variance formula comes directly from the previous step's mean result

Calculating population variance requires obtaining the population mean μ first. In this problem, μ=70 is a necessary input directly substituted into the variance formula.

1. Formula
   Observation

   First displays μ=70, then each term in the σ² formula subtracts 70.

Before calculating population variance, one must know the mean μ, because the deviation is x_i-μ.

1. Formula
   Observation

   The screen first obtains σ²=144, then writes σ=√144=12.

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

Population standard deviation is directly obtained by taking the square root of the population variance.

1. Formula
   Observation

   The example fully applies the three formulas for mean, variance, and standard deviation.

The example is an application of these definitions to specific data.

1. Formula
   Observation

   μ = (67+79+55+61+88)/5 = 70

1. Arithmetic Mean
2. Population Variance

1. Formula
   Observation

   The formula uses 1/n instead of 1/(n-1)

1. The video does not explicitly contrast sample variance; this is a supplementary judgment by the analyst based on the formula form

1. Population Variance

1. Formula
   Observation

   The screen directly provides the population variance formula and calculation.

1. Definition of Population Variance
2. Calculating Population Variance from Definition

1. Formula
   Observation

   The formula uses 1/5, and the problem states there are 5 members.

1. Definition of Population Variance
2. Different Denominators for Population and Sample Variance

1. Formula
   Observation

   The screen displays σ=√144=12.

2. Audio
   Observation

   The narration works through the population weights using the mean, the average squared deviation and the standard deviation in that order.

1. Definition of Population Standard Deviation
2. Finding Population Standard Deviation from Variance

1. Animation
   Observation

   The blanks in the problem are filled with 144 and 12 sequentially.

1. Population Variance and Standard Deviation of Cheerleader Weights
2. Calculating Population Variance from Definition
3. Finding Population Standard Deviation from Variance