中位数(续) . A representative, central number . If data set has a center o Less sensitive to outliers than the average ◆ For skewed data, represents the typical case(代表性个案即 大多数的)” better than the average does g Incomes Average income for a country equally divides the total, which may include some very highincomes a Median income chooses the middle person (halfearn less, halfearn more), giving less influence to high incomes (if any)
中位数（续） ◆ A representative, central number ⚫ If data set has a center ◆ Less sensitive to outliers than the average ◆ For skewed data, represents the “typical case（代表性个案即 大多数的）” better than the average does ⚫ e.g., incomes ◼ Average income for a country equally divides the total, which may include some very high incomes ◼ Median income chooses the middle person (half earn less, half earn more), giving less influence to high incomes (if any)
Example:消费( Spending) Customers plan to spend(Thousands) 3.8.1.4.0.3.0.6.2.8.5.5.0.9.1.1 ◆Rank(秩) ordered from smallest to largest Do 0.3.0.6.0.9.1.1.1.4.2.8.3.8.5.5 R 123415678 Rank of median (1+8)/2=4.5 64 Median is(1.1+1.4)/2=1.25 3188 e Smaller than the average, 2.05 012345 Due to slight skewness? Median verage
Example: 消费（Spending） ◆ Customers plan to spend ($thousands) 3.8, 1.4, 0.3, 0.6, 2.8, 5.5, 0.9, 1.1 ◆ Rank(秩) ordered from smallest to largest 0.3, 0.6, 0.9, 1.1, 1.4, 2.8, 3.8, 5.5 1 2 3 4 5 6 7 8 ◆ Median is (1.1+1.4)/2 = 1.25 ⚫ Smaller than the average, 2.05 ◼ Due to slight skewness? Rank of median = (1+8)/2 = 4.5 0 1 2 3 4 5 3 1 8 8 5 6 4 9 Median Average
Example: The Crash of October 19.1987 The market lost about 20% of its value in one day Dow-Jones Industrials, stock-price changes as each stock began trading that fateful morning ◆ Fairly normal(近似正态) e Mean and median are similar 20% -10% 0% Median =-8.6% Percent change at opening Average=-8.2%
Example: The Crash of October 19,1987 The market lost about 20% of its value in one day ◆ Dow-Jones Industrials, stock-price changes as each stock began trading that fateful morning ◆ Fairly normal（近似正态） ◆ Mean and median are similar 0 5 -20% -10% 0% Percent change at opening Frequency Average = -8.2% Median = -8.6%
EXample: Incomes(Many small values some moderate valuesla few large and very large values) Personal income of 100 people Average is higher than median due to skewness 50 40 30 20 10 0 S100,000 S200.000 Income Average=$38, 710 Median =$27, 216
Example: Incomes(Many small values\some moderate values\a few large and very large values) ◆ Personal income of 100 people ◆ Average is higher than median due to skewness 0 10 20 30 40 50 $0 $100,000 $200,000 Income Average = $38,710 Median = $27,216 Frequency
众数(Mode) e Also summarizes the data e Most common data value o Middle of tallest histogram bar Mode ● Problems: Depends on how you draw histogram(bin width) Might be more than one mode(two tallest bars) ode e Good if most data values arecorrect ● Good for nominal(名义的)data(eg, elections
众数（Mode） ◆ Also summarizes the data ◆ Most common data value ⚫ Middle of tallest histogram bar ◆ Problems: ⚫ Depends on how you draw histogram (bin width) ⚫ Might be more than one mode (two tallest bars) ◆ Good if most data values are “correct” ◆ Good for nominal(名义的) data (e.g., elections) Mode Mode