【統計圖：箱形圖、幹葉圖、直方圖及累積頻數圖】Statistical Diagrams: Box Plots, Stem-and-Leaf, Histograms and Cumulative Frequency
SketchMath 數學團隊 SketchMath Maths Team ｜ 更新 Updated 2026-10-20
箱形圖顯示五數概括：最小值、Q₁、中位數、Q₃、最大值。幹葉圖保留每一個數據。當組距不同時，直方圖用頻數密度 = 頻數 ÷ 組距。累積頻數曲線可由 50%、25%、75% 水平讀出中位數及四分位數。A box plot shows the five-number summary: minimum, Q₁, median, Q₃, maximum. A stem-and-leaf display keeps every value. A histogram uses frequency density = frequency ÷ class width when the widths differ. A cumulative frequency curve gives the median and quartiles by reading across from 50%, 25% and 75%.
拖動 Q₃：箱變長，四分位數間距 Q₃ − 4 增大，中位數及鬚則不動。Q₃ = 9 即例題 1。 Drag Q₃: the box stretches, the IQR = Q₃ − 4 grows, and the median and whiskers stay where they are. Q₃ = 9 is Example 1. 自己試試 → Try it yourself →
統計圖重點 Diagram Facts
五數概括Five-number summarymin, Q₁, median, Q₃, max
四分位數間距Interquartile rangeIQR = Q₃ − Q₁
頻數密度Frequency densityfrequency / class width
圓形圖的角度Pie chart anglefrequency / total × 360°
累積頻數Cumulative frequency對應各組的上組界繪畫Plot against the upper class boundary
When the classes have different widths, the height of a bar is the frequency density = frequency ÷ class width, so the area of each bar is proportional to its frequency.
改變紅色組別的闊度：高度隨之下降，使面積（頻數 60）保持不變。 Change the width of the red class: its height drops so that the area (the frequency 60) stays the same. 自己試試 → Try it yourself →
Plot cumulative frequency against the upper class boundary. Go across from 50% of the total to the curve and down to the axis for the median; use 25% and 75% for Q₁ and Q₃.
移動水平線 p：向右到曲線，再向下到橫軸。在 50 處讀出中位數。 Move the level p: go across to the curve and down to the axis. At 50 you read the median. 自己試試 → Try it yourself →
⚠️ 累積頻數要對應上組界 Plot cumulative frequency at the upper boundary累積頻數要到該組結束才完整，所以要在上組界描點，而不是組中值。The cumulative frequency up to a class is only complete at the end of the class, so plot the point at the upper class boundary, not the mid-value.
⚠️ 組距不等的直方圖 A histogram with unequal classes若各組組距不同，直方圖的高度是頻數密度，而不是頻數。面積才與頻數成正比。If the classes have different widths, the height is the frequency density, not the frequency. The area is proportional to the frequency.