People's Education Press: Junior High School Math, Grade 8, Part B: Arithmetic Fairness and Weighted Wisdom

In the world of data, not all information is inherently equal. When processing scores from 'Example 1: Speech Competition', if we directly sum up the content, ability, and effect scores and divide by 3, this is known as“Arithmetic Fairness”— each dimension has a weight of 1, ensuring fairness. However, in real competition and decision-making, judges often prioritize one particular skill. Introducing different-sized 'weights' then reveals a precise way to reflect reality:“Weighted Wisdom”.

Understanding 'Weight' and Weighted Average

Generally, if $n$ numbers $x_1, x_2, \cdots, x_n$ have respective weights $w_1, w_2, \cdots, w_n$, then:

$\frac{x_1w_1 + x_2w_2 + \cdots + x_nw_n}{w_1 + w_2 + \cdots + w_n}$

is called theWeighted Average. Weight (weight) represents the importance of a data point. The higher the weight, the stronger its influence on the final average (just like a heavier weight on a physical balance pulls the fulcrum closer).

Example 1: Application to Speech Competition Scores

Suppose contestant A scores very high in content but slightly lower in stage effect. If using 'arithmetic mean', they might tie with contestant B, whose scores are average across all categories. But if we assign a weight of 0.5 to 'content' and 0.2 to 'effect', contestant A’s weighted score will surpass due to their core strength. The weighted average truly reflects the specific value orientation in talent selection.

Frequency as Weight: Handling Grouped Data

When analyzing large-scale data (e.g., 'Example 6: Monthly Sales at Department Store' or age survey of divers), the same values appear multiple times. At this point, the number of occurrences (frequency) naturally becomes the weight for that value.

When calculating the average of $n$ numbers, if $x_1$ appears $f_1$ times, $x_2$ appears $f_2$ times, ..., $x_k$ appears $f_k$ times (where $f_1 + f_2 + \cdots + f_k = n$), then the average of these $n$ numbers:

$\bar{x} = \frac{x_1f_1 + x_2f_2 + \cdots + x_kf_k}{n}$

is also called the weighted average of these $k$ numbers, where $f_1, f_2, \cdots, f_k$ are referred to as the weights of $x_1, x_2, \cdots, x_k$. This method filters out the impact of individual extreme sales spikes, accurately reflecting the general performance of most staff, thus enabling the creation of incentive programs that are both challenging and realistic.

The Wisdom of Midpoint Approximation

When data is roughly grouped into different intervals, we lose the exact individual values. At this point, themidpointrefers to the average of the two endpoints of the interval. For example, multiplying the midpoint by the frequency within the interval forms the classic weighted calculation pattern:

$\bar{x} = \frac{11 \times 3 + 31 \times 5 + 51 \times 20 + 71 \times 22 + 91 \times 18 + 111 \times 15}{3 + 5 + 20 + 22 + 18 + 15}$

🎯 Core Principle: Finding the True Center of Data

Whether set intentionally as 'importance' or naturally arising from 'frequency statistics', the essence of weight is to assign corresponding gravitational pull to data. The weighted average is not simple arithmetic division—it helps us find the 'true center' in complex data, one that is not easily misled by outliers.

QUESTION 1

A company seeks to hire a public relations officer and evaluated candidates A and B through interview and written exams. Candidate A scored 86 in the interview and 90 on the written exam; Candidate B scored 92 in the interview and 83 on the written exam. If the company considers the interview score more important than the written exam and assigns weights of 6 and 4 respectively, who will be selected based on the scores?

Candidate A will be selected because their written exam score is higher

Candidate A will be selected, with a weighted average score of 88 points

Candidate B will be selected, with a weighted average score of 88.4 points

Candidate B will be selected, with a weighted average score of 87.5 points

QUESTION 2

Chenguang Middle School sets the semester sports grade out of 100 points, where morning exercise and after-school activities account for 20%, mid-term exam for 30%, and final exam for 50%. Xiao Tong’s three scores are 95, 90, and 85. What is Xiao Tong’s final sports grade for the semester?

88.5 points

90 points

89 points

87.5 points

QUESTION 3

The age distribution among female volleyball players is as follows: 13 years old (1 person), 14 years old (4 people), 15 years old (5 people), 16 years old (2 people). Find the average age of the team (round to the nearest integer).

14 years old

15 years old

16 years old

14.5 years old

QUESTION 4

If a store needs to hire someone responsible for unpacking, assembling, and shelving goods, assigning weights of 2, 3, and 5 to 'Computer Skills', 'Language Ability', and 'Product Knowledge' respectively. Given candidate A’s scores are 60, 70, and 90, and candidate B’s scores are 80, 80, and 70. Based on weighted average scores, who should be hired?

Candidate A, weighted average score of 78 points

Candidate A, weighted average score of 76 points

Candidate B, weighted average score of 77.5 points

Candidate B, weighted average score of 75 points

QUESTION 5

Stores A and B sell identical items at the same price normally. During the Spring Festival, both offer discounts: Store A sells all items at 80% of the original price; Store B offers 70% off only on the amount exceeding 200 yuan per purchase. If the shopping amount is $x$ yuan ($x > 200$), which strategy saves more money?

When $x = 600$, both stores save equally; go to Store B if $x > 600$, go to Store A if $x < 600$

Store A always saves more money

Store B always saves more money

When $x = 400$, both stores save equally

Statistical Challenge: Uncovering the Truth Behind the Data

Apply Weighted Wisdom to Write Realistic Reports and Make Decisions

In real life, whether monitoring vehicle speeds by traffic police, planning inventory for stores, or judging international sports competitions, absolute 'arithmetic averages' often obscure the truth. Only by understanding the patterns of 'weights' and 'frequencies' can we make wise decisions.

Task 1

【Writing Task - Traffic Police Speed Report】
The monitoring system recorded the speed ranges and frequencies of 100 vehicles passing through a junction during a certain period: 41–51 km/h (10 vehicles); 51–61 km/h (30 vehicles); 61–71 km/h (40 vehicles); 71–81 km/h (20 vehicles). Apply statistical knowledge learned (using midpoints and weighted averages) to calculate the average speed and write a brief report to advise traffic officers.

Step-by-step Solution and Report Example:
1. Extract midpoints: The midpoint of 41–51 is 46; 51–61 is 56; 61–71 is 66; 71–81 is 76.
2. Calculate the weighted average:
$\bar{x} = \frac{46 \times 10 + 56 \times 30 + 66 \times 40 + 76 \times 20}{100}$
$\bar{x} = \frac{460 + 1680 + 2640 + 1520}{100} = \frac{6300}{100} = 63$ km/h.

📝 Traffic Police Report Example:
“Reporting to Command Center: A total of 100 vehicles passed through the speed check junction during this period. Using midpoint analysis of grouped data, the weighted average speed on this route is approximately 63 km/h. 40% of vehicles were traveling between 61–71 km/h, representing the main flow speed. Additionally, 20 vehicles reached speeds of 71–81 km/h; it is recommended to enhance speed warning alerts during this period.”

Task 2

【Business Decision - Sportswear Sales】
A store’s pie chart shows the sales distribution of a sportswear item by size: S (24%), M (30%), L (22%), XL (16%), XXL (8%). If next month’s total inventory is 2000 pieces, provide specific procurement recommendations for this store.

Detailed Analysis:
In sales statistics, a pie chart directly displays the frequency distribution of historical sales data by size. These percentages represent customers’ preferences for different sizes—the 'weight'. Since size M has the highest sales frequency (mode effect), procurement strategy should strictly follow this weighted proportion:
1. M size should be the priority: $2000 \times 30\% = 600$ pieces.
2. S and L sizes come next: S size order $2000 \times 24\% = 480$ pieces; L size order $2000 \times 22\% = 440$ pieces.
3. XL and XXL quantities should be reduced: XL needs 320 pieces, XXL only 160 pieces.
Recommendation Summary:Adopt a differentiated procurement strategy, allocating resources according to historical sales proportions as weights, to avoid stockouts or overstocking of unpopular sizes.

Task 3

【Special Weighting - Outlier Removal in Gymnastics and Furniture Factories】
In a gymnastics competition, six judges scored an athlete: 9.4, 8.9, 8.8, 8.9, 8.6, 8.7. The rule requires removing one highest and one lowest score, then calculating the average of the remaining scores.
(1) Calculate the athlete’s final score.
(2) From the perspective of 'weighted average', explain why the rule removes the highest and lowest scores?

Detailed Analysis:
(1) Calculate the score: The dataset is {8.6, 8.7, 8.8, 8.9, 8.9, 9.4}. Remove the highest score 9.4 and the lowest score 8.6. The remaining 4 valid scores are 8.7, 8.8, 8.9, 8.9.
Arithmetic average = $(8.7 + 8.8 + 8.9 + 8.9) \div 4 = 35.3 \div 4 = 8.825$ points.

(2) Weighted Perspective Explanation: Removing the highest and lowest scores is essentially a special weighting strategy: it sets theweight to 0for extreme outliers (possibly due to judge bias or errors), while setting theweight to 1. This forcibly neutralizes the drastic pull of individual outliers on the overall score, ensuring the final average is more stable and fair. In later study of range and variance, you will deeply understand how abnormal fluctuations affect central tendency.