Can the candies in my fist tell me about the whole box? — Foundations of descriptive and inferential statistics

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Can a few candies held in my fist speak for the whole candy box? Using the candy-drawing metaphor, we distinguish descriptive statistics, which summarizes the drawn candies as they are, from inferential statistics, which estimates the whole box from their pattern.

The Week 10 lecture for the graduate course Communication Research Methods II at Hallym University Media School. The two sets of Week 5 materials from the 2024 undergraduate research methods course were merged into one and rebuilt with Achmage Presentation, keeping their wording and order unchanged. It covers population and sample; the mode · median · mean; variance and standard deviation; the normal distribution and the trap of the mean; and the confidence interval obtained with t.test.

The R code on the slides is shown together with the results of actually running it in R 4.3.3, and the results on the exercise slides appear on the next click. The candy-drawing simulator slide links to a page that ports the R Shiny code to the web, where you can stack up samples 10 · 100 · 1000 times and watch the sample proportions gather in a bell shape around the population proportion. The concept diagrams were drawn with Achmage Pele, and the photos and illustrations are originals from the 2024 lecture materials (original sources not stated).

Includes the Korean slides and full reading version, with English introduction and reading.

01. 10W-1

Communication Research Methods II (Fall 2026)

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02. Population and sample – a Halloween random candy and chocolate box

Photo of a Halloween candy box full of candies and chocolates

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03. Let us grab some candy with our fists.

Population

The method does not matter. What are the exact proportions of the candies in the whole box?

Shall we grab just a few?

Sample 1

OR · Shall we grab a fistful so full it bursts?

Sample 2

Photos, from left: a box full of candy (population), a fist and only a few candies (Sample 1), and a fist and a big pile of bubble gum (Sample 2)

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04. Let us grab some candy with our fists.

Population

Candy box – the entire population of Korea

Shall we grab just a few?

Sample 1 – survey of 1,000 people

OR · Shall we grab a fistful so full it bursts?

Sample 2 – survey of 100,000 people

Photos, from left: a box full of candy (population), a fist and only a few candies (Sample 1), and a fist and a big pile of bubble gum (Sample 2)

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05. Population and sample

• Population: the entire group the researcher wants to know about (e.g., “the entire population of Korea”)

• Sample: a subset of the population that the researcher can observe and that carries the population's characteristics (e.g., “Since I cannot know the entire population of Korea… one by one, I picked only 1000 people out of the entire population of Korea… - a sample of 1000”)

• I am not a tycoon, so I cannot survey everyone in Korea (a census of the population is impossible)

• With a mere few million won, I can survey about 1000–2000 people (drawing a sample according to the research purpose)

• What makes it possible to estimate the entire population of Korea from the data of those 1000–2000 people I collected is none other than statistics (estimating the population through a sample)

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06. Descriptive statistics and inferential statistics

Inside the outer box “Population: the entire population of Korea” is “A population within the population: Korea's entire tennis-playing population”. An arrow goes from the tennis-playing population to “A sample of the population within the population: the sample of tennis players I picked to represent Korea's entire tennis-playing population..”. From the sample, the flow leads to “Analyzing the characteristics of the sample” (descriptive statistics: the features of the tennis-player sample I picked), then through “Judging whether to generalize” to “Estimating the characteristics of the whole population” (inferential statistics: whether tennis players will influence the image of Korean tennis), and the last arrow returns to the population.

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07. What is descriptive statistics?

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08. Descriptive statistics

Descriptive Statistics

• A statistical method that identifies the characteristics of data through the organization, presentation, summarization, interpretation, etc. of data collected from measurements or experiments
• A method of organizing the information contained in data through numerical summaries (sum, mean, variance, median, etc.) or graphical summaries of data obtained from the entire population or from a sample (sample: part of the population)
• Explaining the characteristics of data based on their basic statistical information

In the top row, an arrow goes from “Population” to “Parameter”; in the bottom row, from “Sample” to “Sample statistics”. Both arrows are labeled “Descriptive statistics”.

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09. Descriptive statistics

Descriptive Statistics

• Describing the data as they are
• Not a 3-line summary… but (mostly) a one-line summary

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10. Population and sample – candy-drawing simulator

What I drew with my fist came out as nothing but strawberry chocolates – but the box had an even mix?

Overlaid on the candy-drawing simulator screen are ‘Shall we grab just a tiny handful?’ with a fist and a few candies on the right, and a photo of the candy box (population) at the bottom left. Candy-drawing simulator screen: at the top left, a slider for the number to draw and a table of counts for each kind of candy drawn; below, a bar chart by kind

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11. Descriptive statistics

Descriptive Statistics

• Counting how many candies of each kind there are within the bunch of drawn candies,
• Calculating the mean count, median, mode, standard deviation, etc. for each kind of candy,
• Graphing the distribution of each kind of candy, and so on,
• The work of concretely identifying the characteristics within the bunch of drawn candies (the sample)

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12. Types and interpretation of descriptive statistics – what to use for a ‘one-line summary’

Descriptive statistics (expressing the characteristics of data in numbers)

Central Tendency

• Mode
• Median
• Mean

Dispersion

• Range
• Variance
• Standard deviation

Distribution

• Kurtosis
• Skewness

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13. Types and interpretation of descriptive statistics

Descriptive statistics – Central Tendency

Statistics that address which value represents the collected data, or around which value the data are concentrated

Mode	The value with the highest frequency among all the data
Median	The value located exactly in the middle when all the data are arranged in rank order from the lowest value to the highest
Mean	The value obtained by adding up all the data and then dividing the total by the number of data points

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14. Types and interpretation of descriptive statistics

Mean

Table 2-2 Monthly sales and number of purchases for shopping malls A and B

Shopping mall	A	B
Monthly sales	3,000,000 won	3,000,000 won
Monthly purchases	100 purchases	100 purchases
Average amount per purchase	30,000 won	30,000 won

Table 2-3 Monthly sales and number of purchases for shopping malls A and B after free shipping for orders of 35,000 won or more

Shopping mall	A	B
Monthly sales	3,300,000 won	6,000,000 won
Monthly purchases	110 purchases	120 purchases
Average amount per purchase	30,000 won	50,000 won

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15. Why R?

R (Quantitative Finance)

• Software specialized in performing statistical analysis, like SPSS
• With SPSS you pay a very high price every month, but R is open source (free)
• It can run almost all of the most complex statistics performed in SPSS
• The only drawback: R requires a little coding knowledge
• But if you use GPT, even that drawback drops in difficulty to the point where you hardly need to learn R coding at all

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16. Installing R & R-Studio

https://posit.co/download/rstudio-desktop/

RStudio Desktop download page: the two steps ‘1: Install R’ and ‘2: Install RStudio’ with download buttons

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17. Installing R & R-Studio

https://cran.rstudio.com/

CRAN (The Comprehensive R Archive Network) front page: links to download R for Linux · macOS · Windows

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18. Installing R & R-Studio

https://cran.rstudio.com/

The ‘R-4.3.3 for Windows’ page. The ‘Download R-4.3.3 for Windows’ link at the very top is marked with a red circle and ‘Click’

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19. Installing R & R-Studio

https://cran.rstudio.com/

Two R for Windows 4.3.3 setup screens: on the left, the step for accepting the GNU General Public License; on the right, the setup-complete screen

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20. Installing R & R-Studio

https://posit.co/download/rstudio-desktop/

RStudio Desktop download page. The ‘2: Install RStudio’ button is marked with a red circle and ‘Click’

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21. Installing R & R-Studio

https://posit.co/download/rstudio-desktop/

The ‘RStudio Setup’ start screen: the message ‘Starting RStudio Setup’ and the Next button

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22. Running R-Studio

RStudio on first launch. The New File button at the top left is marked with a red circle and ‘Click’

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23. Running R-Studio

RStudio with the New File menu open. The ‘R Script’ item at the very top is marked with a red circle and ‘Click’

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24. Running R-Studio

RStudio with an empty R script pane open

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25. Running R-Studio: Ctrl+C / Ctrl+V the following code as is

student_data <- data.frame(
  name = c("학생1", "학생2", "학생3", "학생4", "학생5", "학생6", "학생7", "학생8",
           "학생9", "학생10", "학생11", "학생12", "학생13", "학생14", "학생15"),
  phone_usage = c(7.2, 5.8, 6.5, 8.1, 4.3, 6.7, 7.9, 5.2, 6.0, 7.5, 5.6, 6.3, 8.5, 5.0, 24.0),
  age = c(20, 21, 19, 22, 20, 21, 20, 19, 22, 21, 20, 23, 19, 21, 20),
  major = c("경영학", "컴퓨터공학", "영문학", "물리학", "화학공학", "통계학", "경제학",
            "정치외교학", "생명과학", "전자공학", "국문학", "수학", "건축공학", "심리학", "사회학"))

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26. Running R-Studio: Ctrl+C / Ctrl+V the following code as is

Select all and click Run

Or Ctrl+Enter

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27. Example data for descriptive statistics

name	phone_usage	age	major
Student 1	7.2	20	Business administration
Student 2	5.8	21	Computer engineering
Student 3	6.5	19	English literature
Student 4	8.1	22	Physics
Student 5	4.3	20	Chemical engineering
Student 6	6.7	21	Statistics
Student 7	7.9	20	Economics
Student 8	5.2	19	Political science and diplomacy

name	phone_usage	age	major
Student 9	6.0	22	Life sciences
Student 10	7.5	21	Electronic engineering
Student 11	5.6	20	Korean literature
Student 12	6.3	23	Mathematics
Student 13	8.5	19	Architectural engineering
Student 14	5.0	21	Psychology
Student 15	24.0	20	Sociology

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28. Types and interpretation of descriptive statistics - mode

Descriptive statistics – Central Tendency

Statistics that address which value represents the collected data, or around which value the data are concentrated

Mode	The value with the highest frequency among all the data
Median	The value located exactly in the middle when all the data are arranged in rank order from the lowest value to the highest
Mean	The value obtained by adding up all the data and then dividing the total by the number of data points

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29. AI & R Exercise - Mode

Prompts:

• “Find the mode of phone_usage in this dataset”
• “To find the mode of phone_usage in this dataset, count each value within that variable”

Check with R code:

freq <- table(student_data$phone_usage)
maxfreq <- max(freq)
which(freq == maxfreq)
table(student_data$phone_usage)

Check it with your own eyes

R output

> which(freq == maxfreq)
4.3   5 5.2 5.6 5.8   6 6.3 6.5 6.7 7.2 7.5 7.9 8.1 
  1   2   3   4   5   6   7   8   9  10  11  12  13 
8.5  24 
 14  15 
> table(student_data$phone_usage)

4.3   5 5.2 5.6 5.8   6 6.3 6.5 6.7 7.2 7.5 7.9 8.1 
  1   1   1   1   1   1   1   1   1   1   1   1   1 
8.5  24 
  1   1

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30. Example data for descriptive statistics – the answer for the mode

name	phone_usage	age	major
Student 1	7.2	20	Business administration
Student 2	5.8	21	Computer engineering
Student 3	6.5	19	English literature
Student 4	8.1	22	Physics
Student 5	4.3	20	Chemical engineering
Student 6	6.7	21	Statistics
Student 7	7.9	20	Economics
Student 8	5.2	19	Political science and diplomacy

name	phone_usage	age	major
Student 9	6.0	22	Life sciences
Student 10	7.5	21	Electronic engineering
Student 11	5.6	20	Korean literature
Student 12	6.3	23	Mathematics
Student 13	8.5	19	Architectural engineering
Student 14	5.0	21	Psychology
Student 15	24.0	20	Sociology

Every value has a count of just 1
So there is no mode

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31. Types and interpretation of descriptive statistics - median

Descriptive statistics – Central Tendency

Statistics that address which value represents the collected data, or around which value the data are concentrated

Mode	The value with the highest frequency among all the data
Median	The value located exactly in the middle when all the data are arranged in rank order from the lowest value to the highest
Mean	The value obtained by adding up all the data and then dividing the total by the number of data points

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32. AI & R Exercise - Median

Prompts:

• “Find the median of phone_usage in this dataset”
• “To find the median of phone_usage in this dataset, arrange each value within that variable in order from smallest to largest”

Check with R code:

median(student_data$phone_usage)

Check it with your own eyes

R output

> median(student_data$phone_usage)
[1] 6.5

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33. Example data for descriptive statistics – the answer for the median

name	phone_usage	age	major
Student 5	4.3	20	Chemical engineering
Student 14	5.0	21	Psychology
Student 8	5.2	19	Political science and diplomacy
Student 11	5.6	20	Korean literature
Student 2	5.8	21	Computer engineering
Student 9	6.0	22	Life sciences
Student 12	6.3	23	Mathematics
Student 3	6.5	19	English literature

name	phone_usage	age	major
Student 6	6.7	21	Statistics
Student 1	7.2	20	Business administration
Student 10	7.5	21	Electronic engineering
Student 7	7.9	20	Economics
Student 4	8.1	22	Physics
Student 13	8.5	19	Architectural engineering
Student 15	24.0	20	Sociology

6.5 is exactly the 8th value whether you count from the top or the bottom
So the median is 6.5

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34. Types and interpretation of descriptive statistics - mean

Descriptive statistics – Central Tendency

Statistics that address which value represents the collected data, or around which value the data are concentrated

Mode	The value with the highest frequency among all the data
Median	The value located exactly in the middle when all the data are arranged in rank order from the lowest value to the highest
Mean	The value obtained by adding up all the data and then dividing the total by the number of data points

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35. AI & R Exercise - Mean

Prompts:

• “Find the mean value of phone_usage in this dataset”
• “Show the calculation steps for finding the mean value of phone_usage in this dataset, and find the mean value”

Check with R code:

mean(student_data$phone_usage)

Check by calculating it yourself on a calculator

R output

> mean(student_data$phone_usage)
[1] 7.64

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36. Example data for descriptive statistics – the answer for the mean value

name	phone_usage	age	major
Student 5	4.3	20	Chemical engineering
Student 14	5.0	21	Psychology
Student 8	5.2	19	Political science and diplomacy
Student 11	5.6	20	Korean literature
Student 2	5.8	21	Computer engineering
Student 9	6.0	22	Life sciences
Student 12	6.3	23	Mathematics
Student 3	6.5	19	English literature

name	phone_usage	age	major
Student 6	6.7	21	Statistics
Student 1	7.2	20	Business administration
Student 10	7.5	21	Electronic engineering
Student 7	7.9	20	Economics
Student 4	8.1	22	Physics
Student 13	8.5	19	Architectural engineering
Student 15	24.0	20	Sociology

Sum of all values: 114.6
The sum divided by 15: 7.64

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37. The trap of the mean

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38. Example data for descriptive statistics – the problem with the mean value: Outlier

name	phone_usage	age	major
Student 5	4.3	20	Chemical engineering
Student 14	5.0	21	Psychology
Student 8	5.2	19	Political science and diplomacy
Student 11	5.6	20	Korean literature
Student 2	5.8	21	Computer engineering
Student 9	6.0	22	Life sciences
Student 12	6.3	23	Mathematics
Student 3	6.5	19	English literature

name	phone_usage	age	major
Student 6	6.7	21	Statistics
Student 1	7.2	20	Business administration
Student 10	7.5	21	Electronic engineering
Student 7	7.9	20	Economics
Student 4	8.1	22	Physics
Student 13	8.5	19	Architectural engineering
Student 15	24.0	20	Sociology

1 student uses a smartphone for 24 hours · What if we leave this student out and recalculate? 90.6 / 14 = 6.47

The students' average smartphone time is 6.5 hours, not 7.6 hours.

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39. R-Studio example 2: Ctrl+C / Ctrl+V the following code as is

student_data <- data.frame(
  name = c("학생1", "학생2", "학생3", "학생4", "학생5", "학생6", "학생7", "학생8",
           "학생9", "학생10", "학생11", "학생12", "학생13", "학생14", "학생15"),
  phone_usage = c(7.2, 5.8, 6.5, 8.1, 4.3, 6.7, 7.9, 5.2, 6.0, 7.5, 5.6, 6.3, 8.5, 5.0, 24.0),
  age = c(20, 21, 19, 22, 20, 21, 20, 19, 22, 21, 20, 23, 19, 21, 20),
  major = c("경영학", "컴퓨터공학", "영문학", "물리학", "화학공학", "통계학", "경제학",
            "정치외교학", "생명과학", "전자공학", "국문학", "수학", "건축공학", "심리학", "사회학"),
  salary = c(rep(5000, 14), 100000)
)

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40. R-Studio example 2: salary

name	phone_usage	age	major	salary
Student 1	7.2	20	Business administration	5000
Student 2	5.8	21	Computer engineering	5000
Student 3	6.5	19	English literature	5000
Student 4	8.1	22	Physics	5000
Student 5	4.3	20	Chemical engineering	5000
Student 6	6.7	21	Statistics	5000
Student 7	7.9	20	Economics	5000
Student 8	5.2	19	Political science and diplomacy	5000

name	phone_usage	age	major	salary
Student 9	6.0	22	Life sciences	5000
Student 10	7.5	21	Electronic engineering	5000
Student 11	5.6	20	Korean literature	5000
Student 12	6.3	23	Mathematics	5000
Student 13	8.5	19	Architectural engineering	5000
Student 14	5.0	21	Psychology	5000
Student 15	24.0	20	Sociology	100000

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41. AI & R Exercise - Mean

Prompts:

• “Find the mean value of salary in this dataset”
• “Show the calculation steps for finding the mean value of salary in this dataset, and find the mean value”

Check with R code:

mean(student_data$salary)

R output

> mean(student_data$salary)
[1] 11333.33

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42. R-Studio example 2: the answer for the average salary

name	phone_usage	age	major	salary
Student 1	7.2	20	Business administration	5000
Student 2	5.8	21	Computer engineering	5000
Student 3	6.5	19	English literature	5000
Student 4	8.1	22	Physics	5000
Student 5	4.3	20	Chemical engineering	5000
Student 6	6.7	21	Statistics	5000
Student 7	7.9	20	Economics	5000
Student 8	5.2	19	Political science and diplomacy	5000

name	phone_usage	age	major	salary
Student 9	6.0	22	Life sciences	5000
Student 10	7.5	21	Electronic engineering	5000
Student 11	5.6	20	Korean literature	5000
Student 12	6.3	23	Mathematics	5000
Student 13	8.5	19	Architectural engineering	5000
Student 14	5.0	21	Psychology	5000
Student 15	24.0	20	Sociology	100000

R-Studio code result: 11333.33
Annual salary per person: 113.33 million won

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43. The trap of the mean

• Ah, the average salary came out to 113.33 million won, so Hallym University students' salaries average over 100 million won! = the trap of the mean
• Because of a single unusual person earning over 1 billion won, the average salary gets inflated
• That ‘unusual person’ throws the statistical results off, and in statistics such an ‘unusual person’ is called an Outlier

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44. Median income in Korea

https://www.index.go.kr/unify/idx-info.do?idxCd=8001

Screen of the ‘Household median income’ table in Korea's National Indicator System: household median income (nominal · real) for 2011–2022, with source · notes

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45. Trends in Korea's standard median income (by household size)

https://www.index.go.kr/unity/potal/main/EachDtlPageDetail.do?idx_cd=2762

Screen of the ‘Standard median income’ table in Korea's National Indicator System: standard median income for households of 1 to 7 persons, 2015–2024, with source · notes

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46. Types and interpretation of descriptive statistics

Variance and standard deviation

• Variance: uses the mean (X̄) as the reference for how spread out the data are
• Standard deviation: a measure used when calculating the proportion of data that fall within a certain distance of the mean

To put it more simply, a map that shows how the overall mean value is spread out when you line up the mean values horizontally from the low ones to the high ones

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47. Types and interpretation of descriptive statistics

The 68-95 rule

Figure 2-21 The 68-95 rule: two bars showing 68% from mean−1SD to mean+1SD and 95% from mean−2SD to mean+2SD

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48. Types and interpretation of descriptive statistics

Kurtosis

Positive kurtosis:
turns out the folks who invested in Bitcoin are crowded tightly together

Negative kurtosis:
turns out lots of folks got burned on Bitcoin and sold at a loss, so there are fewer folks holding coins than last year

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49. Types and interpretation of descriptive statistics

Skewness

Figure 2-25 Tails of a distribution: a symmetric distribution (mean=median=mode), a right-tailed distribution (the mean is greater than the median), and a left-tailed distribution (the mean is less than the median)

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50. R-Studio example 2: Ctrl+C / Ctrl+V the following code as is

# 표준편차 곡선을 그리는 코드
curve(dnorm(x, mean = mean(student_data$salary), sd = sd(student_data$salary)),
      from = min(student_data$salary),
      to = max(student_data$salary),
      xlab = "Salary",
      ylab = "Density",
      main = "연봉 분포와 표준편차 곡선",
      col = "blue",
      lwd = 2)

# 연봉 분포를 세로선으로 표시
abline(v = student_data$salary, col = "red", lty = 2)

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51. R-Studio example 2: distribution of the average salary with a student earning 1 billion won mixed in

R graph ‘Salary distribution and standard deviation curve’: the normal distribution curve peaks on the left and falls off in a long slope to the right. Red dashed vertical lines are drawn at 5000 (14 people) and 100000 (1 person).

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52. R-Studio example 2: Ctrl+C / Ctrl+V the following code as is

# 정규분포를 따르는 salary 값을 생성하는 코드
set.seed(42)  # 랜덤 시드를 설정하여 재현성을 보장합니다.

# 평균과 표준편차 지정
mean_salary <- mean(student_data$salary)
sd_salary <- sd(student_data$salary)

# 정규분포에 따르는 새로운 salary 값 생성
new_salary <- rnorm(n = nrow(student_data), mean = mean_salary, sd = sd_salary)

# 생성한 값들을 데이터프레임에 할당
student_data$salary <- new_salary

# 수정된 salary 값 출력
print(student_data$salary)

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The trap of the mean · 16/18

53. R-Studio example 2: normal distribution plot of students with evenly spread salaries

R output

> print(student_data$salary)
 [1]  44961.428  -2518.089  20240.472
 [4]  26856.753  21249.588   8730.216
 [7]  48409.297   9011.452  60843.036
[10]   9795.026  43340.343  67422.217
[13] -22733.884   4494.953   8063.108

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The trap of the mean · 17/18

54. Summary

Mode	The value with the highest frequency among all the data
Median	The value located exactly in the middle when all the data are arranged in rank order from the lowest value to the highest
Mean	The value obtained by adding up all the data and then dividing the total by the number of data points

Variance and standard deviation: a map that shows how the overall mean value is spread out when you line up the mean values horizontally from the low ones to the high ones

Of the three, the mean value is usually the most common, but there are cases, as with salaries, where the mean value does not apply (the mean value is not a cure-all!)

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The trap of the mean · 18/18

55. What is inferential statistics?

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Inferential statistics · 1/23

56. What is inferential statistics?

Basic procedure of inferential statistics

What is inferential statistics? An analytical method that uses sample data to make statistical inferences about a population

Mean and variance

Sample mean	X̄
Estimate of the population mean	μ = X̄
Sample variance	S²
Estimate of the population variance	σ² = S²/n

Confidence interval

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Inferential statistics · 2/23

57. Population and sample – how many strawberry chocolates are there in the whole box?

Candy-drawing simulator screen, with ‘Shall we grab just a tiny handful?’ and a fist with a few candies at the top right, the candy box (population) at the bottom left, and a photo of strawberry chocolates at the bottom right

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Inferential statistics · 3/23

58. Let us grab some candy with our fists.

Population

The method does not matter. What is the exact proportion of strawberry chocolate candies in the whole box?

Shall we grab just a few?

Sample 1

OR · Shall we grab a fistful so full it bursts?

Sample 2

Photos, from left: a box full of candy (population), a fist and only a few candies (Sample 1), and a fist and a big pile of bubble gum (Sample 2)

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Inferential statistics · 4/23

59. RQ. How many strawberry chocolate candies are there in total among all the original candies?

Halloween candy box

• Population: all the candies in the candy box
• Sample: the candies I drew from the box with my fist

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Inferential statistics · 5/23

60. RQ. How many strawberry chocolate candies are there in total among all the original candies?

Population

I want to know how many “just strawberry chocolates” are in the box

Surely you are not going to tear open this whole box and count them one by one?

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Inferential statistics · 6/23

61. Let us count only the candies in my hand (never mind the original box)

Candy-drawing simulator screen: at the top left, a slider for the number to draw and a table of counts for each kind of candy drawn; below, a bar chart by kind. At the bottom, photos of a fist with a few candies and of strawberry chocolates

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Inferential statistics · 7/23

62. Let us count only the candies in my hand (never mind the original box)

https://posit.cloud/content/5906664

https://posit.cloud/content/7862513

See the candy-drawing simulator R-Studio code.R

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Inferential statistics · 8/23

63. Candy-drawing simulator – try drawing on the web

Run it right in your browser: https://claude.ai/artifact/B3vs3qRXH6HRe755AbKnyh

Candy-drawing simulator web screen: a sample size slider set to 100 and the [Draw a sample] · [Draw 10 · 100 · 1000 times] · [Reset] buttons; in the ‘Cumulative draws’ tab, a histogram of the strawberry chocolate proportions of 1,000 samples (a bell shape centered on 10%, with a dashed normal distribution curve); on the right, the mean · standard deviation of the sample proportions and the proportions for all 10 kinds of candy combined

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Inferential statistics · 9/23

64. Exercise. Let us count the number of candies and the strawberry chocolates with the candy-drawing simulator

	Number of candies	Strawberry chocolates
Sample 1
Sample 2
Sample 3
Sample 4
Sample 5

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Inferential statistics · 10/23

65. RQ. How many strawberry chocolate candies are there in total among all the original candies?

Population

I want to know how many “just strawberry chocolates” are in the box

Inferential statistics: estimating the proportion of strawberry chocolates in the original box from the proportion of strawberry chocolates in the sample

	Number of candies	Strawberry chocolates
Sample 1
Sample 2
Sample 3
Sample 4
Sample 5

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Inferential statistics · 11/23

66. RQ. How many strawberry chocolate candies are there in total among all the original candies?

Population

	Number of candies	Strawberry chocolates
Sample 1
Sample 2
Sample 3
Sample 4
Sample 5

Inferential statistics:

• Estimating the proportion of strawberry chocolates in the original box from the proportion of strawberry chocolates in the sample
• “My fist has this many strawberry chocolates, so judging from that, about how many would there be if I counted all the strawberry chocolates in the original box?” – Estimation

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Inferential statistics · 12/23

67. RQ. How many strawberry chocolate candies are there in total among all the original candies?

Making the strawberry chocolate dataset (R-Studio code)

# 데이터 프레임 생성
candy_data <- data.frame(
  Sample = c('샘플1', '샘플2', '샘플3', '샘플4', '샘플5'),
  Total_Candies = c(100, 100, 100, 100, 100),
  Strawberry_Chocolate = c(8, 7, 12, 9, 9)
)

# 데이터 확인
print(candy_data)

Note: If the numbers you actually drew are different, change the Total and Strawberry numbers

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Inferential statistics · 13/23

68. Finding the descriptive statistics of the strawberry chocolates

#최빈값:
freq <- table(candy_data$Strawberry_Chocolate)
maxfreq <- max(freq)
which(freq == maxfreq)
table(candy_data$Strawberry_Chocolate)
#중앙값:
median(candy_data$Strawberry_Chocolate)
#평균값:
mean(candy_data$Strawberry_Chocolate)
#분산:
var(candy_data$Strawberry_Chocolate)
#표준편차:
sd(candy_data$Strawberry_Chocolate)

Note: candy_data$Strawberry_Chocolate has the structure ‘dataset$variable_name’. For the rest of the descriptive statistics formulas, just use the formulas shown alongside as they are

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Inferential statistics · 14/23

69. Finding the descriptive statistics of the strawberry chocolates – R output

> which(freq == maxfreq)
9 
3 
> table(candy_data$Strawberry_Chocolate)

 7  8  9 12 
 1  1  2  1 
> median(candy_data$Strawberry_Chocolate)
[1] 9
> mean(candy_data$Strawberry_Chocolate)
[1] 9
> var(candy_data$Strawberry_Chocolate)
[1] 3.5
> sd(candy_data$Strawberry_Chocolate)
[1] 1.870829

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Inferential statistics · 15/23

70. Inferential statistics: estimating the number of strawberry chocolates in the whole box

# t.test를 사용하여 신뢰구간 계산
t.test(candy_data$Strawberry_Chocolate)$conf.int

R output

> t.test(candy_data$Strawberry_Chocolate)$conf.int
[1]  6.677059 11.322941
attr(,"conf.level")
[1] 0.95

Note: candy_data$Strawberry_Chocolate has the structure ‘dataset$variable_name’
For the rest of the descriptive statistics formulas, just use the formulas shown alongside as they are

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Inferential statistics · 16/23

71. Inferential statistics: estimating the number of strawberry chocolates in the whole box

# t.test를 사용하여 신뢰구간 계산
t.test(candy_data$Strawberry_Chocolate)$conf.int

t.test result

> t.test(candy_data$Strawberry_Chocolate)$conf.int
[1]  6.677059 11.322941
attr(,"conf.level")
[1] 0.95

In other words, there is a 95% probability that the whole box contains between 6.68 and 11.32 strawberry chocolates

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Inferential statistics · 17/23

72. The beginning of inferential statistics: sampling and estimation

Characteristics of sample quantities

Definition 3-1 Population, sample, parameter, statistic

• 1) Population: the entire group that is the object of interest
• 2) Sample: a subset of the population
• 3) Parameter: a characteristic value of the population (e.g., population mean, population proportion, population variance, etc.)
• 4) Statistic: a characteristic value of the sample (e.g., sample mean, sample proportion, sample variance, etc.)

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Inferential statistics · 18/23

73. Inferential statistics: estimating the population (the candy box) from candy sample statistics

“We ‘estimate’ the population's parameters from the sample's statistics” = “I ‘estimate’ the distribution of candies in the candy box from the 100 candies in my fist”

Definition 3-1 · Population, sample, parameter, statistic	Candy example
Population: the entire group that is the object of interest	(the whole candy box)
Sample: a subset of the population	(the handful I drew with my fist, including strawberry chocolates)
Parameter: a characteristic value of the population (e.g., population mean, population proportion, population variance, etc.)	(the mean, frequency, etc. of each candy in the whole candy box)
Statistic: a characteristic value of the sample (e.g., sample mean, sample proportion, sample variance, etc.)	(the mean, frequency, etc. of each candy in my fist)

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Inferential statistics · 19/23

74. Premise 1 of inferential statistics: the population cannot be known ‘exactly’

“We ‘estimate’ the population's parameters from the sample's statistics” = “I ‘estimate’ the distribution of candies in the candy box from the 100 candies in my fist”

“I ‘estimate’ the distribution of candies in the candy box from the 100 candies in my fist”
= in other words, it is impossible to know ‘exactly how many’ strawberry chocolates are in the candy box
(one of the most important premises of statistics)

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Inferential statistics · 20/23

75. Premise 2 of inferential statistics: there is always a chance of being wrong

# t.test를 사용하여 신뢰구간 계산
t.test(candy_data$Strawberry_Chocolate)$conf.int

t.test result

> t.test(candy_data$Strawberry_Chocolate)$conf.int
[1]  6.677059 11.322941
attr(,"conf.level")
[1] 0.95

In other words, there is a 95% probability that the whole box contains between 6.68 and 11.32 strawberry chocolates

→ There is a 5% chance that I could be wrong (there is no such thing as a zero chance of being wrong)

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Inferential statistics · 21/23

76. Premise 3 of inferential statistics: a ‘standard error’ exists (a number that shows how wrong I am)

“We ‘estimate’ the population's parameters from the sample's statistics” = “I ‘estimate’ the distribution of candies in the candy box from the 100 candies in my fist”

(e.g., the number of mint candies in the population estimated from the 100 candies in my fist = 3.4 billion)

• Huh, the actual number is 3.59 billion, but it does not match the 3.4 billion I predicted?
• Standard error: the degree of “how wrong I was” expressed as a number (the degree of difference between the population parameter and the parameter estimated from the sample's descriptive statistics)

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Inferential statistics · 22/23

77. Summary of the basics of inferential statistics

• It is impossible to know perfectly a specific parameter of the population (e.g., the total number of strawberry chocolates in the box)
• Searching through the entire population (inside the box) is a brute-force and inefficient method
• I count only the candies in my palm, then “estimate” the original box from the “pattern” of the candies in my palm
• We cannot know the perfect answer because we “estimate” (we guess using probability)
• “(Since I cannot guess the population perfectly) how wrong I am” = the standard error.

Population

I want to know how many “just strawberry chocolates” are in the box

Surely you are not going to tear open this whole box and count them one by one?

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Inferential statistics · 23/23