> 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/2-4.-relative-position-of-data.md). # 2-4. Relative Position of Data 数据的相对位置 ## 1. Percentiles and Quartiles > *Given an observed value* $$x$$ *in a data set,* $$x$$ *is the* **Pth percentile** *of the data if the percentage of the data that are less than or equal to* $$x$$ *is P. The number P is the* **percentile rank** *of* $$x$$ *.* **EXAMPLE 27.** 다음의 데이터 세트에 대하여 1.39의 percentile을 구하라. 3.33의 percentile도 구하라. ``` 1.90 3.00 2.53 3.71 2.12 1.76 2.71 1.39 4.00 3.33 ``` **\[Solution]** {% 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) length(x[x <= 1.39]) / length(x) * 100 # x[x<=1.39] : elements of x less than or equal to 1.39 length(x[x <= 3.33]) / length(x) * 100 # x[x<=3.33] : elements of x less than or equal to 3.33 ``` {% endtab %} {% tab title="Result 1" %} ``` > length(x[x <= 1.39]) / length(x) * 100 ## [1] 10 ``` {% endtab %} {% tab title="Result 2" %} ``` > length(x[x <= 3.33]) / length(x) * 100 ## [1] 80 ``` {% endtab %} {% endtabs %} The $$P$$th percentile cuts the data set in two so that approximately $$P$$ % of the data lie below it and $$(100−P)%$$ of the data lie above it. In particular, the three percentiles that cut the data into fourths, as shown in Figure "Data Division by Quartiles", are called the **quartiles**. The following simple computational definition of the three quartiles works well in practice. \[ **Data Division by Quartiles** ] ![Data Division by Quartiles](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-LqsT56UBUm4xsHBsjZp%2F-LqsTHHciX02xvAg-_Cv%2F20191011_102919.png?alt=media\&token=92ec8fb0-c644-4da2-aa79-d08beb28235f) > For any data set: > > 1. The **second quartile** **Q2** of the data set is its median. > 2. Define two subsets: > 3. the **lower set**: all observations that are strictly less than **Q2**; > 4. the **upper set**: all observations that are strictly greater than **Q2**. > > 3\. The **first quartile** **Q1** of the data set is the median of the lower set. **EXAMPLE 28.** 앞의 예에 있는 데이터 세트의 quartiles를 구하라. **\[Solution]** {% 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) # 1. quantile y <- quantile(x) y # quartiles of x y[3] # second quartile of x # 2. lower subset sort(x[xy[3]]) # 4. 5%-, 10%-, 90%-, 95%-th quantile # 1) 5%th quantile quantile(x, probs=0.05) # 2) 10%-th quantile quantile(x, probs=0.10) # 3) 5%-, 10%-, 90%-, 95%-th quantile quantile(x, probs = c(0.05, 0.10, 0.90, 0.95)) ``` {% endtab %} {% tab title="1. quartiles" %} ``` > # 1. quantile > y <- quantile(x) > y # quartiles of x ## 0% 25% 50% 75% 100% ## 1.3900 1.9550 2.6200 3.2475 4.0000 > y[3] # second quartile of x ## 50% ## 2.62 ``` {% endtab %} {% tab title="2. lower subset" %} ``` > # 2. lower subset > sort(x[x # 3. upper subset > sort(x[x>y[3]]) ## [1] 2.71 3.00 3.33 3.71 4.00 ``` {% endtab %} {% tab title="4. %th quantile" %} ``` > # 4. 5%-, 10%-, 90%-, 95%-th quantile > # 1) 5%th quantile > quantile(x, probs=0.05) ## 5% ## 1.5565 > > # 2) 10%-th quantile > quantile(x, probs=0.10) ## 10% ## 1.723 > > # 3) 5%-, 10%-, 90%-, 95%-th quantile > quantile(x, probs = c(0.05, 0.10, 0.90, 0.95)) ## 5% 10% 90% 95% ## 1.5565 1.7230 3.7390 3.8695 ``` {% endtab %} {% endtabs %} **EXAMPLE 29.** 앞의 데이터 세트에 3.88을 추가하고 quartiles를 구하라. **\[Solution]** {% 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, 3.88) # 1. quantile y <- quantile(x) y # quartiles of x y[3] # second quartile of x # 2. lower subset sort(x[xy[3]]) # 4. 5%-, 10%-, 90%-, 95%-th quantile # 1) 5%th quantile quantile(x, probs=0.05) # 2) 10%-th quantile quantile(x, probs=0.10) # 3) 5%-, 10%-, 90%-, 95%-th quantile quantile(x, probs = c(0.05, 0.10, 0.90, 0.95)) ``` {% endtab %} {% tab title="1. quartiles" %} ``` > # 1. quantile > y <- quantile(x) > y # quartiles of x ## 0% 25% 50% 75% 100% ## 1.39 2.01 2.71 3.52 4.00 > y[3] # second quartile of x ## 50% ## 2.71 ``` {% endtab %} {% tab title="2. lower subset" %} ``` > # 2. lower subset > sort(x[x # 3. upper subset > sort(x[x>y[3]]) ## [1] 3.00 3.33 3.71 3.88 4.00 ``` {% endtab %} {% tab title="4. %th quantile" %} ``` > # 4. 5%-, 10%-, 90%-, 95%-th quantile > # 1) 5%th quantile > quantile(x, probs=0.05) ## 5% ## 1.575 > > # 2) 10%-th quantile > quantile(x, probs=0.10) ## 10% ## 1.76 > > # 3) 5%-, 10%-, 90%-, 95%-th quantile > quantile(x, probs = c(0.05, 0.10, 0.90, 0.95)) ## 5% 10% 90% 95% ## 1.575 1.760 3.880 3.940 ``` {% endtab %} {% endtabs %} **Quartile vs Quantile vs Percentile** | **Quartile** | Quantile | Percentile | | ------------------- | --------------- | ---------------- | | 0 Quartile | = 0 Quantile | = 0 Percentile | | 1 Quartile (**Q1**) | = 0.25 Quantile | = 25 Percentile | | 2 Quartile (**Q2**) | = 0.5 Quantile | = 50 Percentile | | 3 Quartile (**Q3**) | = 0.75 Quantile | = 75 Percentile | | 4 Quartile | = 1 Quantile | = 100 Percentile | * **Percentiles** go from 0 to 100**.** * **Quartiles** go from 1 to 4 (or 0 to 4). * **Quantiles** can go from anything to anything. * **Percentiles and quartiles** are examples of **quantiles**. ### **1.1 Five-Number summary** of the data set: ``` {xmin, Q1, Q2, Q3, xmax} ``` * The five-number summary is used to construct a **box plot.** ![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-LqsWvSU2xEkOKRHPhVe%2F-LqsXG1uy35rneZHehys%2F20191011_105716.png?alt=media\&token=0b955699-b369-430a-b328-cc3f86fa22cf) ### ### 1.2 Five Numbers in R **EXAMPLE 30. EXAMPLE 28.**의 five-numbers를 구하라. **\[ Solution ]** {% 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) fvn_x <- fivenum(x); fvn_x ``` {% endtab %} {% tab title="Five Numbers" %} ``` > fvn_x <- fivenum(x); fvn_x ## [1] 1.39 1.90 2.62 3.33 4.00 ``` {% endtab %} {% endtabs %} ## 2. IQR > **The interquartile range (IQR)** is the quantity : > > $$IQR = Q3−Q1$$ **EXAMPLE 31.** **Example 28.**의 Box Plot을 작성하라. 그리고 IQR을 구하라. **\[Solution]** {% 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) # 1. Box Plot boxplot(x) # 2. IQR quantile(x) IQR(x) ``` {% endtab %} {% tab title="1. Box Plot" %} ![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lqt3dPZKfJL7hC8I4A9%2F-Lqt4AO4LpvJKGUUM5XW%2Fimage.png?alt=media\&token=e7711f53-c311-4810-9537-b81009882820) {% endtab %} {% tab title="2. Quantile and IQR" %} ``` # 2. IQR > quantile(x) ## 0% 25% 50% 75% 100% ## 1.3900 1.9550 2.6200 3.2475 4.0000 > IQR(x) ## [1] 1.2925 ``` {% endtab %} {% endtabs %} ## 3. Box Plot **EXAMPLE 32.** Draw the Box plot of **Example 28**. {% 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) # change the plot window size : (7×7) ⇒ (7×4) win.graph(4, 7) # 1. horizontal box plot : horizontal=TRUE boxplot(x, main="Box Plot", col="cyan") # 2. display the points ⇒ points() len <- length(x) points(rep(1,len), x) # position of points : (x, 1) # 3. computing fivenumbers ⇒ fivenum() ⇒ (min, Q1, Q2, Q3, max) xfn = fivenum(x); xfn # display fivenumbers (min, Q1, Q2, Q3, max) ⇒ text(x, y, labels) text(0.7, xfn, labels=xfn, pos=3, cex=0.8) # 4. display legend (Mean, Std Dev., n) ⇒ legend() # paste0() legend("topright", paste0(c("Mean", "Std Dev.", "n"), "=", round(c(mean(x), sd(x), length(x)), 3)), text.col = 4, cex=0.8) ``` {% endtab %} {% tab title="1. Box Plot" %} ![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-LrPx3SB5zjhzGRLcQHS%2F-LrPxNof4IFzOUnx4fWy%2Fimage.png?alt=media\&token=ec3021f1-c975-4435-bb8a-8ef16d196f05) {% endtab %} {% tab title="2. Points" %} ![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-LrPx3SB5zjhzGRLcQHS%2F-LrPxSGQAEt79ReAkIze%2Fimage.png?alt=media\&token=ae8ec40f-53b0-450c-a8d7-692a752f1457) {% endtab %} {% tab title="3. Five Numbers" %} ![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-LrPx3SB5zjhzGRLcQHS%2F-LrPxY1nKAF4JgIoxPr0%2Fimage.png?alt=media\&token=36021c25-4790-47df-8c42-e91a6f810029) {% endtab %} {% tab title="4. Legend" %} ![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-LrPx3SB5zjhzGRLcQHS%2F-LrPxb8mERNYuOj_axe5%2Fimage.png?alt=media\&token=7ff43851-8183-4aea-b408-e3a286b710aa) {% endtab %} {% endtabs %} **EXAMPLE 33.** Horizontal Box Plot {% 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) # change the plot window size : (7×7) ⇒ (7×4) win.graph(7, 4) # 1. horizontal box plot : horizontal=TRUE boxplot(x, horizontal=T, main="Box Plot", col="cyan") # 2. display the points ⇒ points() len <- length(x) # the number of data points(x, rep(1,len)) # position of points : (x, y=1) # 3. computing fivenumbers ⇒ fivenum() ⇒ (min, Q1, Q2, Q3, max) xfn <- fivenum(x); xfn # display fivenumbers (min, Q1, Q2, Q3, max) ⇒ text(x, y, labels) text(xfn, 0.65, labels=xfn, pos=3, cex=0.8) # cex=0.8 : size of tex # pos = 1, 2, 3, 4 # 4. display legend (Mean, Std Dev., n) ⇒ legend() # paste0() legend("topright", paste0(c("Mean", "Std Dev.", "n"), "=", round(c(mean(x), sd(x), length(x)), 3)), text.col = 4, # color of legend text cex=0.8) # size of legend text ``` {% endtab %} {% tab title="1. Horizontal Box Plot" %} ![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-LrPu-i5Z-wufgeYYEfz%2F-LrPu9R1nWY5PYMkW_bu%2Fimage.png?alt=media\&token=a4b55f4f-595a-4e03-a7d1-1bef64e66e8e) {% endtab %} {% tab title="2. Points" %} ![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-LrPu-i5Z-wufgeYYEfz%2F-LrPuCnUzthWXwq0yf3i%2Fimage.png?alt=media\&token=1d8e85d0-5b46-43f1-8515-9d7a79481069) {% endtab %} {% tab title="3. FIve Numbers" %} ![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-LrPu-i5Z-wufgeYYEfz%2F-LrPuHpAkQDy7Cigz7GX%2Fimage.png?alt=media\&token=06b26400-cfb8-4d3f-af16-87baae021e2c) {% endtab %} {% tab title="4. Legend" %} ![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-LrPu-i5Z-wufgeYYEfz%2F-LrPv5dFz9nCAEJ9Wl1i%2Fimage.png?alt=media\&token=acaf0112-cf5a-47b0-992d-a87ef3c71669) {% endtab %} {% endtabs %} **EXAMPLE 34.** (Two Factor Box plot) 다음의 두 개의 데이터 세트를 하나의 Box-plot으로 표시하라. ``` 11분반 성적 : 42.5, 25, 37.5, 40, 27.5, 30, 35, 40, 25, 42.5, 37.5, 52.5, 37.5, 60, 65, 30, 32.5, 57.5, 32.5, 50, 55, 55, 40, 32.5, 35, 47.5 12분반 성적 : 32.5, 52.5, 40, 42.5, 12.5, 57.5, 67.5, 32.5, 57.5, 35, 45, 40, 72.5, 45, 47.5, 70, 45, 42.5, 50, 40, 57.5, 42.5, 57.5, 42.5, 35, 35, 32.5, 25, 67.5, 57.5, 37.5, 25 ``` **\[ Solution ]** {% tabs %} {% tab title="R Source" %} ``` x <- c(42.5, 25, 37.5, 40, 27.5, 30, 35, 40, 25, 42.5, 37.5, 52.5, 37.5, 60, 65, 30, 32.5, 57.5, 32.5, 50, 55, 55, 40, 32.5, 35, 47.5) y <- c(32.5, 52.5, 40, 42.5, 12.5, 57.5, 67.5, 32.5, 57.5, 35, 45, 40, 72.5, 45, 47.5, 70, 45, 42.5, 50, 40, 57.5, 42.5, 57.5, 42.5, 35, 35, 32.5, 25, 67.5, 57.5, 37.5, 25) boxnames <- c("11분반", "12분반") boxplot(x, y, main="2019학년도 중간고사 성적", names=boxnames, ylim=c(0, 80), notch=TRUE) # 2. display the points ⇒ points() # 11분반 len <- length(x) points(rep(1,len), x) # position of points : (x, 1) # 12분반 len <- length(y) points(rep(2,len), y) # position of points : (x, 1) # 3. computing fivenumbers ⇒ fivenum() ⇒ (min, Q1, Q2, Q3, max) # display fivenumbers (min, Q1, Q2, Q3, max) ⇒ text(x, y, labels) text(0.7, fivenum(x), labels=fivenum(x), pos=3, cex=0.8) text(1.7, fivenum(y), labels=fivenum(y), pos=3, cex=0.8) # 4. display legend (Mean, Std Dev., n) ⇒ legend() # paste0() legend("bottomleft", paste0(c("Mean", "Std Dev.", "n"), "=", round(c(mean(x), sd(x), length(x)), 3)), text.col = 4, cex=0.8) legend("bottomright", paste0(c("Mean", "Std Dev.", "n"), "=", ``` {% endtab %} {% tab title="Box Plot" %} ![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-LsS2XxL671EJ6YAIG2t%2F-LsS2_wMJ9WEwPNJtaL0%2Fimage.png?alt=media\&token=bfd8b26a-72b2-4b3a-a3ff-32b7af1b9035) {% endtab %} {% endtabs %} ## 4. Find out outliers > $$\[ Q1 - 1.5 \* IQR, Q3 + 1.5 \*IQR]$$의 범위를 벗어나는 데이터를 **outlier**라 한다. **EXAMPLE 35.** 다음의 데이터 세트에서 이상치(outlier)를 찾아내라. ``` 1.90 3.00 2.53 3.71 2.12 1.76 2.71 1.39 4.00 3.33 3.88 7.30 ``` **\[ Solution 1 ]** {% 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, 3.88, 7.30) y <- quantile(x); names(y) <- NULL y1 <- y[2] - 1.5 * IQR(x); y1 # y[2] = Q1, lower limit y2 <- y[4] + 1.5 * IQR(x); y2 # y[4] = Q3, uypper limit which(x < y1) # find out the index of x which is less than y1 which(x > y2) # find out the index of x which is greater than y2 ``` {% endtab %} {% tab title="Find out the Index of x" %} ``` > y1 <- y[2] - 1.5 * IQR(x); y1 # lower limit ## [1] -0.46625 > y2 <- y[4] + 1.5 * IQR(x); y2 # upper limit ## [1] 6.28375 > > which(x < y1) ## integer(0) > which(x > y2) ## [1] 12 ``` There is one outlier. The index of outlier is `12`. -> x\[12] {% endtab %} {% tab title="Outlier" %} ``` > x[12] ## [1] 7.3 ``` {% endtab %} {% endtabs %} **\[ Solution 2 ]** Using `boxplot()`, you can also find out outliers... {% 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, 3.88, 7.30) xbox <- boxplot(x) xbox ``` {% endtab %} {% tab title="Outliers" %} ``` > xbox ## $stats <- lower whisker : 1.390, lower hinge : 2.010, median : 2.855, upper hinge : 3.795, upper whisker : 4.000 [,1] ## [1,] 1.390 ## [2,] 2.010 ## [3,] 2.855 ## [4,] 3.795 ## [5,] 4.000 $n ## [1] 12 $conf ## [,1] ## [1,] 2.04085 ## [2,] 3.66915 $out <- outlier ## [1] 7.3 $group ## [1] 1 $names ## [1] "1" > ``` {% endtab %} {% tab title="Box Plot with Outlier" %} ![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-LsRxKNsx2OPhHJjNX9V%2F-LsRxO_Dv-sdPrPQLaiM%2Fimage.png?alt=media\&token=e590419a-c4f9-4ffc-b818-e2003d28f193) {% endtab %} {% tab title="Box Plot without Outlier" %} ``` boxplot(x,horizontal=TRUE,outline=FALSE) # outline=FALSE ``` ![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-LsRxKNsx2OPhHJjNX9V%2F-LsRxipx8heMdxABeahY%2Fimage.png?alt=media\&token=e713a166-c45c-458e-b22c-60d6689642a2) {% endtab %} {% endtabs %} 참고자료 : * [박스플롯에 대하여](https://boxnwhis.kr/2019/02/19/boxplot.html) * Lower whisker and Upper Whisker : min and max * [The Lower Hinge and Upper Hinge](https://www.statisticshowto.datasciencecentral.com/upper-hinge-lower-hinge/) ## 5. z-Score > The **z-score** of an observation $$x$$ is the number of $$z$$ given by the computational formula > > $$z = (x - \bar{x}) / {\sigma}$$ or $$z = (x - \mu) / \sigma$$ > > according to whether the data set is a sample or is the entire population. ![x-scale vs z-score](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-LqtBj2XnT6ajA8HAKGs%2F-LqtCJUK2pbNhTDJhJoy%2Fimage.png?alt=media\&token=8aed60c5-7712-4371-b122-b756d3140683) **EXAMPLE 36.** 다음과 같은 10명의 학생의 평균 평점에 대하여 z-score를 구하라. ``` 1.90 3.00 2.53 3.71 2.12 1.76 2.71 1.39 4.00 3.33 ``` **\[Solution]** {% 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) z_score <- (x - mean(x)) / sd(x) round(z_score, 2) ``` {% endtab %} {% tab title="z-score" %} ``` > round(z_score, 2) ## [1] -0.86 0.41 -0.13 1.23 -0.61 -1.02 0.07 -1.45 1.56 0.79 ``` {% endtab %} {% endtabs %} **EXAMPLE 37.** 최근에 등록했던 학생들 평균 평점의 평균과 표준편차가 각각 $$\mu = 2.70$$과 $$\sigma = 0.50$$ 이었다. Antonio와 Beatrice 두 학생의 z-score는 각각 -0.62와 1.28이었다. 그들의 평균 평점은 몇 점인가? **\[Solution]** {% tabs %} {% tab title="R Source" %} ``` mean <- 2.70 std <- 0.50 z_score <- c(-0.62, 1.28) GPA <- mean + std * z_score ``` {% endtab %} {% tab title="z-score" %} ``` > GPA <- mean + std * z_score; GPA ## [1] 2.39 3.34 ``` {% endtab %} {% endtabs %} 1. 数据的相对位置 1\) 百分位数(percentile) 2\) 分位数(quantile) 3\) 四分位距(IRQ) 4\) 箱形图(Box Plot) 5\) Outlier 6\) Five-Numbers Summary