Ch 3 Central Tendency 中央集中趨勢測量

Central Tendency中央趨勢 資料分配的中心位置 中央趨勢的主要統計值 平均值Mean ( ) 中位數Median 眾數Mode

Mean 平均數 The most commonly used measure of central tendency. Formula: or Used for interval, ratio variables (primarily) and ordinal variables (sometimes) 母體平均值 µ 樣本平均值

Mean (continued) 平均數特質 平均數為資料的平衡點 各觀察值與平均值的差的總合等於零 各觀察值與平均值的差的平方和為最小 優點: 使用到每個觀察值來決定資料的中心位置 缺點:易受極端值的影響

Median 中位數(Me) 依數值大小順序排列的觀察值中位居中央者 The score at the exact center of a distribution Used for either ordinal or interval,ratio variables 特質 不受極端值的影響 不易進行代數演算 不易進行統計推論

Mode 眾數 觀察值中出現最多的一個數值 The score that occurs most frequently Used for nominal variables Used when have a skewed distribution (有極端值時) 特質: 不受極端值的影響 可能有多個或沒有 對觀察值的個數或數值變化感應不靈敏 it can not appropriately represent the “center” of a distribution

Median vs. Mean Median is less sensitive to the extreme scores Mean can be affected by the extreme scores mean increases when we have more positive extreme scores (positive skew) mean decreases when we have more negative extreme scores (negative skew) When and only when a distribution is symmetrical, the two means will be the same.

Measures of Position 資料位置的測量

Measures of Position Three common measures of position Percentile 百分位數 Deciles 十分位數 Quartiles 四分位數

Quartiles Quartiles divide the distribution into four quarters. The first quartile (Q1)=25%, the second quartile (Q2) =50%…and so forth Q2=??

Deciles Deciles divide the distribution of scores into tenths. The first decile = 10%, the second decile=20% …..and so forth 5th decile=??

Percentile Used to determine the location of a score or/ to identify the value of a given percentile. 求第k 個百分位數 (Pk)

9支股票本益比 17 18.5 20 25 36 45 57 68 93 計算 Q1 Q2 Q3 D6 P75

15名隨機抽取的學生成績 62 72 39 73 55 90 58 96 83 60 74 76 81 81 72 畫出分數枝葉圖 Mean Median Q1 Q3 D60 P85