Slide 1 — MGS 2150 Business Statistics

第1页——MGS 2150 商业统计学

Knowledge Points (知识点)

Course: MGS 2150 Business Statistics (商业统计学)
Chapter 3: Numerical Measures (数值度量)
Focus: Measures of Location & Variability (位置与变异度量)

Explanation (解释)

Business Statistics focuses on using numerical and statistical tools to analyze, summarize, and interpret business data for better decision-making.
商业统计学专注于运用数值和统计工具来分析、总结并解释商业数据，以帮助做出更好的决策。

Example (例子)

A company records monthly sales for one year. By applying numerical measures, managers can calculate:

Mean (平均值): average sales
Median (中位数): middle sales value
Range (极差): highest – lowest sales

例如，一家公司记录了一年的月销售额。通过数值度量，管理者可以计算：

平均值：平均销售额
中位数：销售额的中间值
极差：最高与最低销售额的差

Extension (拓展)

Statistical analysis is critical in business areas like:

Finance: measuring stock returns variability
Operations: tracking variability in production times
Marketing: analyzing consumer spending patterns

统计分析在商业中的应用：

金融：测量股票收益的波动性
运营：跟踪生产时间的变异性
市场：分析消费者的消费模式

Summary (总结)

本页介绍了课程和主题，强调数值度量在商业决策中的作用。
This slide introduces the course and highlights the role of numerical measures in decision-making.

Slide 2 — Measures of Location and Variability

第2页——位置度量与变异度量

Knowledge Points (知识点)

Measures of Location (位置度量): focus on central tendency
Measures of Variability (变异度量): focus on dispersion

Explanation (解释)

Location measures (like mean, median) tell us where the data is centered.
位置度量（如均值、中位数）告诉我们数据“集中在哪里”。

Variability measures (like range, variance, standard deviation) show how spread out the data is.
变异度量（如极差、方差、标准差）展示了数据“分散程度”。

Example (例子)

Comparing two products’ battery life:

Product A: Average = 3 hrs, Range = (2, 4)
Product B: Average = 3 hrs, Range = (2.5, 3.5)
→ Both have the same average, but variability is different.

比较两个产品的电池续航：

产品A：平均 = 3 小时，范围 = 2–4
产品B：平均 = 3 小时，范围 = 2.5–3.5
→ 两者平均相同，但离散度不同。

Extension (拓展)

Managers consider both average performance and stability when making decisions. A product with stable performance may be preferred even if its average is slightly lower.
管理者在决策时既考虑 平均表现，也考虑 稳定性。即使平均表现稍低，性能稳定的产品更可能被选择。

Summary (总结)

位置度量 + 变异度量 = 数据的完整画像。
Location and variability together provide a full picture of data.

Slide 3 — Range (极差)

第3页——极差

Knowledge Points (知识点)

Range = Max – Min
Simplest measure of variability, but highly sensitive to outliers

Explanation (解释)

Range is calculated as the difference between the largest and smallest values in the dataset.
极差通过数据中最大值与最小值的差来计算。

It’s simple but not robust, because one extreme value can distort the measure.
它计算简单，但不稳定，因为一个极端值可能会使结果失真。

Example (例子)

Apartment rents:

Max = 615, Min = 425
Range = 615 – 425 = 190

公寓租金：

最大值 = 615，最小值 = 425
极差 = 190

Extension (拓展)

Finance: daily range of stock prices (High – Low) shows market volatility.
Operations: range of production times indicates process consistency.

在金融中，股票的“日内波动范围”体现市场波动；在生产中，工序时间的极差说明一致性。

Summary (总结)

极差简单直观，但容易受极端值影响。
The range is easy to compute but unstable due to sensitivity to outliers.

Slide 4 — Interquartile Range (四分位距)

第4页——四分位距

Knowledge Points (知识点)

IQR = Q3 – Q1
Focuses on middle 50% of data
Less sensitive to outliers

Explanation (解释)

The interquartile range (IQR) measures the spread of the middle 50% of data. It removes the impact of extreme values by ignoring the top 25% and bottom 25%.
四分位距（IQR）衡量数据中间50%的离散程度。它忽略顶部25%和底部25%的极端值，因此更稳健。

Example (例子)

Apartment rents:

Q1 = 445, Q3 = 525
IQR = 525 – 445 = 80

公寓租金：

第一四分位数 = 445，第三四分位数 = 525
IQR = 80

Extension (拓展)

IQR is widely used in outlier detection. Data points outside the range:

Q 1-1.5 \times I QR, Q 3 + 1.5 \times I QR

are considered potential outliers.

IQR 广泛用于 异常值检测。如果数据点落在：

Q 1-1.5 \times I QR, Q 3 + 1.5 \times I QR

之外，就可能是异常值。

Summary (总结)

IQR 克服了极差对异常值的敏感性，更适合真实数据分析。
IQR reduces sensitivity to outliers, making it more robust than range.

Slide 5 — Variance (方差)

第5页——方差

Knowledge Points (知识点)

Variance measures average squared deviation from the mean
Uses all data points, not just extremes
Different formulas for sample and population

Explanation (解释)

Variance calculates the squared difference between each observation and the mean, then takes the average.
方差通过计算每个观测值与均值的差的平方，并取平均值来衡量数据的变异性。

Sample variance (样本方差):

s^{2} = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n - 1}

Population variance (总体方差):

σ^{2} = \frac{\sum ( x _{i} - μ ) ^{2}}{N}

Example (例子)

If three exam scores are: 80, 85, 95

Mean = 86.67
Deviations: -6.67, -1.67, 8.33
Squared deviations = 44.5, 2.8, 69.4
Variance = (44.5 + 2.8 + 69.4) / (3 – 1) = 58.35

三个考试成绩：80, 85, 95

均值 = 86.67
偏差：-6.67, -1.67, 8.33
偏差平方：44.5, 2.8, 69.4
方差 = (44.5+2.8+69.4) / (3–1) = 58.35

Extension (拓展)

Variance allows comparison of datasets with the same range but different distributions.
In finance, variance measures risk of an investment portfolio.

方差可以比较极差相同但分布不同的数据集。在金融中，方差衡量投资组合的风险。

Summary (总结)

方差利用所有数据点来衡量波动性，是比极差和IQR更全面的指标。
Variance gives a complete measure of variability using all data values.

Slide 6 — Variance (Formula and Concept)

第6页——方差公式与概念

Knowledge Points (知识点)

Variance uses all data values
Measures squared deviations from the mean
Sample vs. population formulas differ

Explanation (解释)

Variance is the average squared deviation from the mean.
方差是所有数据点与均值差的平方的平均值。

Sample variance (样本方差):
$s^{2} = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n - 1}$
Population variance (总体方差):
$σ^{2} = \frac{\sum ( x _{i} - μ ) ^{2}}{N}$

Example (例子)

Dataset = {2, 4, 6}

Mean:
$\overset{x}{ˉ} = \frac{2 + 4 + 6}{3} = 4$
Variance:
$s^{2} = \frac{( 2 - 4 ) ^{2} + ( 4 - 4 ) ^{2} + ( 6 - 4 ) ^{2}}{3 - 1} = \frac{4 + 0 + 4}{2} = 4$

Extension (拓展)

Variance ≥ 0 (always non-negative).
Units are squared, making interpretation harder.
More stable than range or IQR, because it considers all data.

Summary (总结)

方差是基础的波动性度量，能够全面反映数据离散程度。

Slide 7 — Standard Deviation (标准差)

第7页——标准差

Knowledge Points (知识点)

SD = positive square root of variance
Same units as data
Easier to interpret than variance

Explanation (解释)

Standard deviation is the square root of variance.
标准差是方差的平方根。

Sample SD:
$s = s^{2}$
Population SD:
$σ = σ^{2}$

Example (例子)

Apartment rents:

Variance = 2996.16
Standard deviation:
$s = 2996.16 \approx 54.74$

Extension (拓展)

SD is the most widely used measure of variability.
In finance: SD = volatility of returns.
In manufacturing: smaller SD = better quality control.

Summary (总结)

标准差比方差更直观，能更清晰反映数据波动。

Slide 8 — Coefficient of Variation (变异系数, CV)

第8页——变异系数 (CV)

Knowledge Points (知识点)

CV = (SD ÷ Mean) × 100%
Measures relative variability
Useful for cross-comparison

Explanation (解释)

Coefficient of variation (CV) shows the size of SD relative to the mean.
变异系数衡量标准差相对于均值的大小。

Sample CV:
$C V = \frac{s}{x ˉ} \times 100%$
Population CV:
$C V = \frac{σ}{μ} \times 100%$

Example (例子)

Apartment rents:

Mean = 490.80
SD = 54.74
CV:
$C V = \frac{54.74}{490.80} \times 100% \approx 11.15%$

Extension (拓展)

CV enables comparison across different units (e.g., RMB vs. USD).
Higher CV = higher relative risk or instability.

Summary (总结)

CV 揭示了“相对波动性”，便于跨数据集比较。

Slide 9 — Example: Apartment Rents Analysis

第9页——例子：公寓租金分析

Knowledge Points (知识点)

Variance = 2996.16
SD = 54.74
CV = 11.15%

Explanation (解释)

Apartment rents dataset summary:

Range:
$615 - 425 = 190$
IQR:
$Q_{3} - Q_{1} = 525 - 445 = 80$
Variance:
$s^{2} = 2996.16$
Standard Deviation:
$s = 54.74$
Coefficient of Variation:
$C V = 11.15%$

Example (例子)

Interpretation:

SD ≈ 55 → typical deviation from mean rent.
CV = 11.15% → moderate variability compared to mean (490.80).

Extension (拓展)

If another city has mean = 1200, SD = 200:

CV:
$\frac{200}{1200} \times 100% = 16.7%$
→ That city’s rents are relatively more volatile.

Summary (总结)

通过方差、标准差和CV结合，可以清晰解读租金的绝对与相对波动性。

Slide 10 — Excel Computation of Variability

第10页——Excel 中的变异性计算

Knowledge Points (知识点)

Excel functions: AVERAGE, MEDIAN, MODE.SNGL, VAR.S, STDEV.S
CV formula = STDEV/AVERAGE × 100%

Explanation (解释)

Excel provides formulas to compute descriptive statistics directly.
Excel 提供了内置函数，方便快速计算描述性统计量。

Example (例子)

Apartment rent dataset (B2:B71):

Mean:
$AVERAGE (B 2 : B 71) = 490.80$
Median:
$MEDIAN (B 2 : B 71) = 475$
Mode:
$MODE.SNGL (B 2 : B 71) = 450$
Variance:
$VAR.S (B 2 : B 71) = 2996.16$
Std. Dev.:
$STDEV.S (B 2 : B 71) = 54.74$
CV:
$\frac{STDEV.S ( B 2 : B 71 )}{AVERAGE ( B 2 : B 71 )} \times 100% = 11.15%$

Extension (拓展)

Excel Data Analysis ToolPak → Descriptive Statistics
→ Provides mean, variance, skewness, kurtosis, etc.

Excel 的数据分析插件可一键生成完整统计表，包括均值、方差、偏度和峰度。

Summary (总结)

Excel 是高效的工具，可以快速得出关键统计量。

Slide 11 — Excel Descriptive Statistics

第11页——Excel 描述性统计结果

Knowledge Points (知识点)

Excel can generate a full descriptive statistics table
Includes mean, standard error, median, mode, variance, skewness, kurtosis, etc.

Explanation (解释)

Descriptive statistics summarize dataset characteristics in one table.
描述性统计表将数据的核心特征汇总在一张表里。

Key metrics:

Mean (均值)
Standard Error (标准误差)
Median (中位数)
Mode (众数)
Variance (方差)
Standard Deviation (标准差)
Range, Min, Max
Skewness (偏度)
Kurtosis (峰度)

Example (例子)

Apartment rents (N = 70):

Mean:
$\overset{x}{ˉ} = 490.8$
Standard Error:
$SE = 6.54$
Median:
$Median = 475$
Mode:
$Mode = 450$
Variance:
$s^{2} = 2996.16$
Standard Deviation:
$s = 54.74$
Range:
$615 - 425 = 190$

Extension (拓展)

Skewness = 0.92 → data is right-skewed (longer right tail).
Kurtosis = -0.33 → flatter distribution than normal.
Managers use these indicators to judge rent distribution shape and risk.

Summary (总结)

Excel 的描述性统计表为数据分布提供全面信息，不仅有集中趋势，还包括偏度和峰度。

Slide 12 — Application of Location & Variability

第12页——位置与变异度量的应用

Knowledge Points (知识点)

Combine central tendency + variability for decision-making
Helps evaluate performance and stability

Explanation (解释)

Location measures describe the “typical” value, variability measures describe the “stability” of data. Both are necessary for sound analysis.
位置度量描述“典型值”，变异度量描述“稳定性”。两者结合才能全面分析。

Example (例子)

Battery life comparison:

Product A:
$\overset{x}{ˉ} = 3 hrs, R an g e = (2, 4)$
Product B:
$\overset{x}{ˉ} = 3 hrs, R an g e = (2.5, 3.5)$

→ Same mean, but B is more consistent.

Extension (拓展)

In business:
- Finance: choose investments with balance between average returns and volatility.
- Operations: assess whether machines deliver stable quality.

在商业中：

金融：在平均回报与风险之间取平衡。
运营：评估机器是否提供稳定的质量。

Summary (总结)

结合均值与波动性，才能做出可靠的商业决策。

Slide 13 — Range vs. IQR vs. Variance/SD

第13页——极差 vs 四分位距 vs 方差/标准差

Knowledge Points (知识点)

Range: simplest, sensitive to outliers
IQR: robust, ignores extreme 25% tails
Variance/SD: comprehensive, considers all data

Explanation (解释)

Each variability measure has its strengths and weaknesses:

Range: quick but unstable
IQR: better at handling outliers
Variance/SD: best overall, but less intuitive (variance units are squared)

Example (例子)

Apartment rents:

Range = 190
IQR = 80
SD = 54.74

Interpretation: SD shows overall deviation, IQR highlights central stability.

Extension (拓展)

Data analysis workflow: usually start with range, then refine with IQR and SD.
Practical use: IQR for detecting outliers, SD for comparing consistency across datasets.

Summary (总结)

极差、IQR 和标准差各有用途，结合使用能全面刻画数据波动。

Slide 14 — Advanced Measures (Skewness & Kurtosis)

第14页——高级度量：偏度与峰度

Knowledge Points (知识点)

Skewness: measures asymmetry
Kurtosis: measures “peakedness” or tails

Explanation (解释)

Skewness (偏度):
$Skewness > 0 ⟹ Right-skewed$
$Skewness < 0 ⟹ Left-skewed$
Kurtosis (峰度):
$Kurtosis = 0 ⟹ Normal distribution$
$Kurtosis > 0 ⟹ Leptokurtic (sharper peak)$
$Kurtosis < 0 ⟹ Platykurtic (flatter)$

Example (例子)

Apartment rents:

Skewness = 0.92 → right-skewed, more high-rent outliers.
Kurtosis = -0.33 → flatter than normal curve.

Extension (拓展)

Finance: skewness signals asymmetry in returns, kurtosis signals tail risk.
Risk management: firms must consider not just mean & SD, but also tail risks.

Summary (总结)

偏度揭示分布的不对称性，峰度揭示尾部与峰值特征。

Slide 15 — Summary of Measures of Location & Variability

第15页——位置与变异度量总结

Knowledge Points (知识点)

Location: Mean, Median, Mode
Variability: Range, IQR, Variance, SD, CV
Shape: Skewness, Kurtosis

Explanation (解释)

A complete descriptive analysis involves:

Central tendency → typical value
Variability → spread of data
Shape → asymmetry & tails

完整的描述性分析包括：

集中趋势 → 典型值
离散程度 → 数据分布的宽度
分布形态 → 不对称性和尾部情况

Example (例子)

Apartment rents illustrate all measures:

Mean = 490.8
Median = 475
Mode = 450
Range = 190
IQR = 80
Variance = 2996.16
SD = 54.74
CV = 11.15%
Skewness = 0.92
Kurtosis = -0.33

Extension (拓展)

For business: combining all measures helps in risk assessment, decision-making, and strategy design.
For research: descriptive statistics are the foundation before hypothesis testing or regression.

Summary (总结)

完整的数据分析 = 集中趋势 + 离散度量 + 分布形态。

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课堂笔记 （数值度量）

Slide 1 — MGS 2150 Business Statistics

第1页——MGS 2150 商业统计学

Knowledge Points (知识点)

Explanation (解释)

Example (例子)

Extension (拓展)

Summary (总结)

Slide 2 — Measures of Location and Variability

第2页——位置度量与变异度量

Knowledge Points (知识点)

Explanation (解释)

Example (例子)

Extension (拓展)

Summary (总结)

Slide 3 — Range (极差)

第3页——极差

Knowledge Points (知识点)

Explanation (解释)

Example (例子)

Extension (拓展)

Summary (总结)

Slide 4 — Interquartile Range (四分位距)

第4页——四分位距

Knowledge Points (知识点)

Explanation (解释)

Example (例子)

Extension (拓展)

Summary (总结)

Slide 5 — Variance (方差)

第5页——方差

Knowledge Points (知识点)

Explanation (解释)

Example (例子)

Extension (拓展)

Summary (总结)

Slide 6 — Variance (Formula and Concept)

第6页——方差公式与概念

Knowledge Points (知识点)

Explanation (解释)

Example (例子)

Extension (拓展)

Summary (总结)

Slide 7 — Standard Deviation (标准差)

第7页——标准差

Knowledge Points (知识点)

Explanation (解释)

Example (例子)

Extension (拓展)

Summary (总结)

Slide 8 — Coefficient of Variation (变异系数, CV)

第8页——变异系数 (CV)

Knowledge Points (知识点)

Explanation (解释)

Example (例子)

Extension (拓展)

Summary (总结)

Slide 9 — Example: Apartment Rents Analysis

第9页——例子：公寓租金分析

Knowledge Points (知识点)

Explanation (解释)

Example (例子)

Extension (拓展)

Summary (总结)

Slide 10 — Excel Computation of Variability

第10页——Excel 中的变异性计算

Knowledge Points (知识点)

Explanation (解释)

Example (例子)

Extension (拓展)

Summary (总结)

Slide 11 — Excel Descriptive Statistics

第11页——Excel 描述性统计结果

Knowledge Points (知识点)

Explanation (解释)

Example (例子)

Extension (拓展)

Summary (总结)

课堂笔记（数值度量）