【PEP】Junior High School Math, Grade 7, Semester 2: Comprehensive or Sampling? — The First Step in Statistical Investigation

Statistics is the science of collecting, organizing, and analyzing data to make inferences and decisions. Like tasting a pot of eight-treasure rice pudding, you don’t need to drink the entire pot to know its sweetness or saltiness—just stir it well and take a spoonful to ‘see the whole picture from a single spot.’ This is the charm of statistical investigation.

Core Concepts: Who Is Our Main Focus?

Before any investigation, we must clearly define our research subjects:

Population (Population): The entire group under examination.
Individual: Each individual object that makes up the population.
Sample (Sample): A subset of objects drawn from the population.
Sample Size (Sample Size): The number of individualsin the sample(Note: It is a number without units).

Choosing the Investigation Method

Why not always conductcomprehensive surveys(surveys that examine every object)?

Scenario A: Census

For example, the sixth national census in 2010. Extremely high accuracy is required, and the data impacts national economy and people's livelihoods—every individual must be accounted for.

Scenario B: Impact Resistance Test

If testing the impact resistance of a batch of cars, a comprehensive survey would mean destroying all new vehicles. In this case,sampling surveys(selecting a portion of objects for investigation and inferring about the whole) is the only viable option.

The Science and Pitfalls of Sampling

To ensure a single spoonful represents the entire pot, we must followsimple random samplingprinciples so that each individual has an equal chance of being selected. We must avoid the following three pitfalls:

Too small: The sample size is too small, making results prone to chance and unable to objectively reflect the population.
Too large: It defeats the purpose of saving time and effort.
Bias: For example, estimating the entire school’s characteristics based only on classmates nearby—this sample lacks representativeness.

🎯 Core Logic

The core of sampling surveys lies in using sample data to infer the overall situation. Its logical formula is: $q \approx \frac{p}{n} \times m$, where $q$ is the estimated value of the population.

QUESTION 1

Which of the following investigations are suitable for comprehensive surveys, and which for sampling surveys? (1) Testing impact resistance of a batch of cars (2) Understanding the heights of students in a class (3) Surveying viewership ratings of the Spring Festival Gala (4) Selecting the fastest sprinter from a school to compete at the city level

(1)(3) Sampling, (2)(4) Comprehensive

(1)(2) Comprehensive, (3)(4) Sampling

(1) Comprehensive, (2)(3)(4) Sampling

All should be comprehensive surveys

QUESTION 2

To estimate the average height of all students in the school, Xiao Ming surveyed only 3 classmates sitting next to him, using their average height as the estimate. Which statement about this survey is correct?

This is a comprehensive survey

The result accurately reflects the population because it’s simple to operate

This is a sampling survey, but the sample size is too small and not representative

The sample size is the total number of students in the school

QUESTION 3

In the 'Beans in a Bottle' experiment, 50 beans were first marked and returned. Then, 100 beans were randomly grabbed, of which 5 had marks. Estimate how many beans are in the bottle.

500 beans

250 beans

1000 beans

150 beans

QUESTION 4

Which of the following statements is correct?

Sample size refers to the names of individuals in the sample

The advantage of sampling surveys is saving time, effort, and minimal destructiveness

To understand nationwide middle school students’ vision, a comprehensive survey should be used

Comprehensive surveys have no error, while sampling surveys inevitably have large errors

QUESTION 5

Xiao Ming wants to understand household education spending in a residential community by surveying 30 students living there. Is this plan reasonable?

Reasonable—30 people is already quite a lot

Unreasonable—the respondents are limited to student families, lacking diversity

Reasonable—all families in the same neighborhood have similar circumstances

Unreasonable—the entire community’s households should be surveyed

QUESTION 6

In statistics, what do we call each individual object that makes up the population?

Sample

Individual

Sample size

Element

QUESTION 7

What method should be used to investigate whether a certain food product’s pigment content meets national standards?

comprehensive surveys

sampling surveys

QUESTION 8

The prerequisite for simple random sampling is:

The sample size must be greater than 100

Each individual has an equal chance of being selected

Investigators can select individuals based on personal preference

Can only be conducted within a small area

QUESTION 9

To understand the weight status of 400 seventh-grade students at a school, 50 students were selected. What does the number 50 represent in this context?

Individual

Sample

Sample size

Population