> For the complete documentation index, see [llms.txt](https://kmis.gitbook.io/statistics/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://kmis.gitbook.io/statistics/ch_02_descriptive_statistics/2-3.-measures-of-variability.md). # 2-3. Measures of Variability (Statistical Dispersion) 다음의 두 데이터 세트가 있다. 각각의 데이터 세트를 점으로 표현한 것이 dot plot이다. | Data Set 1 | 40 | 38 | 42 | 40 | 39 | 39 | 43 | 40 | 39 | 40 | | ---------- | -- | -- | -- | -- | -- | -- | -- | -- | -- | -- | | Data Set 2 | 46 | 37 | 40 | 33 | 42 | 36 | 40 | 47 | 34 | 45 | {% tabs %} {% tab title="R Source" %} ``` library(ggplot2) x1 <- c(40, 38, 42, 40, 39, 39, 43, 40, 39, 40) x2 <- c(46, 37, 40, 33, 42, 36, 40, 47, 34, 45) x <- data.frame(x1, x2) ggplot(x, aes(x = x1)) + geom_dotplot() ggplot(x, aes(x = x2)) + geom_dotplot() ``` {% endtab %} {% tab title="Result 1" %} ![](/files/-LqsKdGAYdJAv7QlTFtB) {% endtab %} {% tab title="Result 2" %} ![](/files/-LqsKlgctqdxP-2DmsQs) {% endtab %} {% endtabs %} 참고사이트 : ## 1. The Range > *The* **range** *of a data set is the number* *R* *defined by the formula*\ > \ > $$R=x\_{max}−x\_{min}$$ > > *where* $$x\_{max}$$ *is the largest measurement in the data set and* $$x\_{min}$$ *is the smallest.* **EXAMPLE 23.** 앞의 2개의 데이터 세트의 range를 구하라. **\[Solution]** {% tabs %} {% tab title="R Source" %} ``` x1 <- c(40, 38, 42, 40, 39, 39, 43, 40, 39, 40) x2 <- c(46, 37, 40, 33, 42, 36, 40, 47, 34, 45) # range of x1 # 1) range_x1 <- max(x1) - min(x1); range_x1 # 2) range(x1) # R Function : range() diff(range(x1)) # range of x2 # 1) range_x2 <- max(x2) - min(x2); range_x2 # 2) range(x2) diff(range(x2)) ``` {% endtab %} {% tab title="Result" %} ``` > # range of x1 > # 1) > range_x1 <- max(x1) - min(x1); range_x1 ## [1] 5 > > # 2) > range(x1) # R Function : range() ## [1] 38 43 > diff(range(x1)) ## [1] 5 > > # range of x2 > # 1) > range_x2 <- max(x2) - min(x2); range_x2 ## [1] 14 > > # 2) > range(x2) ## [1] 33 47 > diff(range(x2)) ## [1] 14 ``` {% endtab %} {% endtabs %} **Note :** `range( )`function of R returns **the minimum value and the maxim value** of the data set. ## 2. The Variance and The Standard Deviation **EXAMPLE 24.** 위의 예에서 **Data Set 2**의 sample variance와 sample standard deviation을 구하라. **\[Solution]** {% tabs %} {% tab title="R Source" %} ``` x <- c(46, 37, 40, 33, 42, 36, 40, 47, 34, 45) # 1. Variance n <- length(x); n y <- (x - mean(x)); y var_x <- sum(y^2)/(n-1); var_x # 2. R Function for Variance : var() var(x) # 3. R Function for Standard Deviation : sd() sd(x) ``` {% endtab %} {% tab title="Variance" %} ``` > # 1. Variance > n <- length(x); n ## [1] 10 > y <- (x - mean(x)); y ## [1] 6 -3 0 -7 2 -4 0 7 -6 5 > var_x <- sum(y^2)/(n-1); var_x ## [1] 24.88889 > > # 2. R Function for Variance : var() > var(x) ## [1] 24.88889 ``` {% endtab %} {% tab title="Standard Deviation" %} ``` > # 3. R Function for Standard Deviation : sd() > sd(x) ## [1] 4.988877 ``` {% endtab %} {% endtabs %} **Note :** In R, `var()` returns the sample variance, i.e. the denominator used in var() function is `(n-1)`. > *The* **sample variance** *of a set of* $$n$$ *sample data is the number* $$s^2$$ *defined by the formula* > > $$s^2= \frac{Σ(x−\bar{x})^2} {n−1}$$ > > *which by algebra is equivalent to the formula* > > $$s^2= \frac{Σx^2−\frac{1}{n}(Σx)^2} {n−1}$$ > > *The* sample **standard deviation** *of a set of* $$n$$ *sample data is the square root of the sample variance, hence is the number* $$s$$ *given by the formulas* > > $$s= \sqrt { \frac{Σ(x−\bar{x})^2} {n−1} } = \sqrt{ \frac{Σx^2−\frac{1}{n}(Σx)^2} {n−1} }$$
**EXAMPLE 25.** 무작위로 선발한 10명의 학생의 평균 평점은 다음과 같다. sample variance와 sample standard deviation을 구하라. ``` 1.90 3.00 2.53 3.71 2.12 1.76 2.71 1.39 4.00 3.33 ``` **\[풀이]** {% tabs %} {% tab title="R Source" %} ``` x <- c(1.90, 3.00, 2.53, 3.71, 2.12, 1.76, 2.71, 1.39, 4.00, 3.33) var(x) sd(x) ``` {% endtab %} {% tab title="Result" %} ``` > var(x) ## [1] 0.7524278 > sd(x) ## [1] 0.8674259 ``` {% endtab %} {% endtabs %} > *The* **population variance** *and* population **standard deviation** *of a set of* $$N$$ *population data are the numbers* $$σ^2$$ *and* $$σ$$ *defined by the formulas* > > $$σ^2=\frac{Σ(x−μ)^2}{N}$$  and  $$σ= \sqrt { \frac{Σ(x−μ)^2}{N} }$$ **\[ Difference between Two Data Sets ]** ![](/files/-LqsKyWmO21rCDCJOkHo) * **See :** [**Degree of Dispersion**](https://rfriend.tistory.com/122) ## 3. Coefficient of Variation In probability theory and statistics, the **coefficient of variation** (**CV**), also known as **relative standard deviation** (**RSD**), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often expressed as a percentage, and is defined as the ratio of the standard deviation $$\sigma$$ to the mean $$\mu$$ (or its absolute value, $$|\mu|$$ ). > 평균에 대한 상대적인 변동성의 크기를 설명할 때에 **변동계수(Coefficient of Variation)**을 사용한다. > > $$CV = \frac {s} {\bar{x}}$$ > > 평균에 대한 표준편차의 비율로 표현된다. > > 변동계수가 클수록, 즉 표준편차가 표본평균에 비해 클수록 자료의 퍼짐진 정도가 더 크다고 할 수 있다. **EXAMPLE 26.** Example 25.의 CV를 구하라. **\[ Solution ]** {% tabs %} {% tab title="R Source" %} ``` # install.packages("goeveg") library(goeveg) x <- c(1.90, 3.00, 2.53, 3.71, 2.12, 1.76, 2.71, 1.39, 4.00, 3.33) # 1. Calculation of CV cv_x <- sd(x) / mean(x); cv_x # 2. R Function : cv() in 'goeveg' package cv(x) ``` {% endtab %} {% tab title="CV" %} ``` > # 1. Calculation of CV > cv_x <- sd(x) / mean(x); cv_x ## [1] 0.3279493 > > # 2. R Function : cv() in 'goeveg' package > cv(x) ## [1] 0.3279493 ``` {% endtab %} {% endtabs %} 1. 离散程度的统计量 1\) 全距(range) 2\) 标准差(standard deviation) 3\) 方差(variance) 4\) 变异系数(coefficient of variation)