📚 IB Mathematics: Effects of Multiplying or Adding a Constant on Statistical Measures | IB数学：数据乘以常数或加常数后的统计量变换
When a data set is transformed by adding a constant to every value or multiplying every value by a constant, the summary statistics change in predictable and systematic ways. Understanding these transformations is essential for IB Mathematics, as they appear in statistics, probability, and modelling contexts.
当一组数据中的每一个值都加上一个常数或乘以一个常数时，数据的汇总统计量会以可预测且有规律的方式发生变化。理解这些变换对于IB数学至关重要，因为它们在统计、概率和建模问题中经常出现。
1. The Basic Idea of Data Transformations | 数据变换的基本概念
A data transformation changes every original value \(x\) into a new value \(y\). The two most important single transformations are \(y = x + b\) and \(y = ax\). These are often combined into the linear transformation \(y = ax + b\).
数据变换将原始数据中的每一个值 \(x\) 变为新的值 \(y\)。两种最重要的单一变换是 \(y = x + b\) 和 \(y = ax\)。它们通常结合为线性变换 \(y = ax + b\)。
Adding a constant \(b\) shifts the entire data set horizontally on the number line. It does not change the spacing between data points. Multiplying by a constant \(a\) rescales the data, changing the spacing by a factor of \(|a|\).
加上常数 \(b\) 会将整个数据集在数轴上水平移动，但不会改变数据点之间的间隔。乘以常数 \(a\) 会对数据进行缩放，使数据点之间的间隔变为原来的 \(|a|\) 倍。
It is important to distinguish between the two cases: a shift affects only measures of location, while a scale change affects both location and dispersion measures.
必须区分这两种情况：平移只影响位置度量，而缩放同时影响位置度量和离散程度度量。
2. Adding a Constant: Effects on Mean, Median, and Mode | 加上常数：对均值、中位数和众数的影响
Suppose every value in a data set is increased by a constant \(b\). Let the original mean be \(\bar{x}\), the original median be \(m\), and the original mode be \(M\). After the transformation \(y = x + b\), the new mean, median, and mode are simply each increased by \(b\).
假设数据集中的每一个值都增加一个常数 \(b\)。设原始均值为 \(\bar{x}\)，原始中位数为 \(m\)，原始众数为 \(M\)。经过变换 \(y = x + b\) 后，新的均值、中位数和众数都分别增加 \(b\)。
\bar{y} = \bar{x} + b, \quad \text{median}(y) = \text{median}(x) + b, \quad \text{mode}(y) = \text{mode}(x) + b
This happens because all data points move by the same amount, so the entire distribution is translated along the number line without changing its shape.
这是因为所有数据点移动了相同的量，因此整个分布沿数轴平移而不改变其形状。
For example, if the mean salary in a company is 50000 and every employee receives a 2000 bonus, the new mean salary is 52000. The median and mode also increase by 2000.
例如，如果某公司员工的平均工资为50000，且每位员工获得2000奖金，则新的平均工资为52000。中位数和众数也同样增加2000。
3. Adding a Constant: No Effect on Spread Measures | 加上常数：对离散程度度量无影响
Measures of spread such as the range, interquartile range (IQR), variance, and standard deviation measure the variability or spacing of data values. Adding a constant shifts every value but does not change any spacing.
极差、四分位距（IQR）、方差和标准差等离散程度度量衡量的是数据的变异性或数据值之间的间隔。加上常数会使所有值移动，但不会改变任何间隔。
\text{Range}(y) = \text{Range}(x), \quad \text{IQR}(y) = \text{IQR}(x)
\text{Var}(y) = \text{Var}(x), \quad \text{SD}(y) = \text{SD}(x)
Consider the data set 2, 4, 6, 8. Its range is 6. If we add 10 to each value, the new data set is 12, 14, 16, 18, and the range is again 6.
考虑数据集 2, 4, 6, 8，其极差为6。如果每个值都加上10，新数据集为 12, 14, 16, 18，极差仍然是6。
This is a useful property: adding a constant changes the location of the data but not its variability. In IB exams, you may be asked to state this directly or to use it in a multiple-step problem.
这是一个有用的性质：加上常数会改变数据的位置，但不改变其变异性。在IB考试中，可能会要求你直接说明这一点，或在多步问题中加以运用。
4. Multiplying by a Constant: Effects on Mean, Median, and Mode | 乘以常数：对均值、中位数和众数的影响
If every value is multiplied by a constant \(a\), then the mean, median, and mode are each multiplied by \(a\). This is true whether \(a\) is positive or negative, but the interpretation differs for negative values.
如果每个值都乘以常数 \(a\)，则均值、中位数和众数都乘以 \(a\)。无论 \(a\) 为正还是负，这一结论都成立，但负值时的解释有所不同。
\bar{y} = a\bar{x}, \quad \text{median}(y) = a \cdot \text{median}(x), \quad \text{mode}(y) = a \cdot \text{mode}(x)
For instance, if the original mean is 10 and we multiply all data by 3, the new mean is 30. If we multiply by −2, the new mean is −20.
例如，如果原始均值为10，将所有数据乘以3，新的均值为30。如果乘以−2，新的均值为−20。
Notice that when \(a\) is negative, the order of the data is reversed. The smallest original value becomes the largest new value, and vice versa. This affects how we interpret the median and the sign of measures of skewness, but the multiplication rule for the median still holds.
注意当 \(a\) 为负数时，数据的顺序会反转。原始的最小值变成新的最大值，反之亦然。这会影响我们对中位数的解释以及偏度度量的符号，但中位数的乘法规则仍然成立。
5. Multiplying by a Constant: Effects on Range and IQR | 乘以常数：对极差和四分位距的影响
Unlike adding a constant, multiplying by a constant changes the spread of the data. The range and IQR are multiplied by the absolute value \(|a|\).
与加上常数不同，乘以常数会改变数据的离散程度。极差和四分位距要乘以 \(|a|\)。
\text{Range}(y) = |a| \cdot \text{Range}(x), \quad \text{IQR}(y) = |a| \cdot \text{IQR}(x)
We use the absolute value because range and IQR are distances, and distances cannot be negative. For example, if the original range is 8 and we multiply the data by 5, the new range is 40. If we multiply by −3, the new range is 24, not −24.
我们使用绝对值，因为极差和四分位距是距离，距离不能为负。例如，如果原始极差为8，将数据乘以5，新的极差为40。如果乘以−3，新的极差为24，而不是−24。
This property is intuitive: multiplying by \(a\) stretches or compresses the number line by factor \(|a|\), so all gaps between data points are scaled by the same factor.
这个性质很直观：乘以 \(a\) 会将数轴拉伸或压缩 \(|a|\) 倍，因此数据点之间的所有间隔都按相同的因子缩放。
6. Multiplying by a Constant: Effects on Variance and Standard Deviation | 乘以常数：对方差和标准差的影响
Standard deviation is a measure of spread in the same units as the data. Therefore, it is multiplied by \(|a|\) when the data are multiplied by \(a\). Since variance is the square of the standard deviation, variance is multiplied by \(a^2\).
标准差是数据同单位的离散程度度量。因此，当数据乘以 \(a\) 时，标准差乘以 \(|a|\)。由于方差是标准差的平方，所以方差乘以 \(a^2\)。
\text{SD}(y) = |a| \cdot \text{SD}(x), \quad \text{Var}(y) = a^2 \cdot \text{Var}(x)
For example, if the original variance is 4 and the data are multiplied by 3, the new variance is \(9 \times 4 = 36\). The new standard deviation is \(3 \times 2 = 6\), which matches \(\sqrt{36} = 6\).
例如，如果原始方差为4，数据乘以3，则新方差为 \(9 \times 4 = 36\)。新的标准差为 \(3 \times 2 = 6\)，这与 \(\sqrt{36} = 6\) 一致。
Notice that even if \(a\) is negative, the variance is multiplied by \(a^2\), which is always non-negative. This is consistent with the fact that variance cannot be negative.
注意即使 \(a\) 为负数，方差乘以 \(a^2\)，而 \(a^2\) 始终非负。这与方差不能为负的事实一致。
7. Combined Transformation y = ax + b | 组合变换 y = ax + b
In IB problems, the transformation is often written as \(y = ax + b\). This combines a scale change by \(a\) and a shift by \(b\). The order matters: first multiply by \(a\), then add \(b\).
在IB问题中，变换通常写成 \(y = ax + b\)。这结合了按 \(a\) 的缩放和按 \(b\) 的平移。顺序很重要：先乘以 \(a\)，然后加上 \(b\)。
The general rules for measures of location are:
位置度量的一般规则如下：
\bar{y} = a\bar{x} + b, \quad \text{median}(y) = a \cdot \text{median}(x) + b, \quad \text{mode}(y) = a \cdot \text{mode}(x) + b
The general rules for measures of spread are:
离散程度度量的一般规则如下：
\text{Range}(y) = |a| \cdot \text{Range}(x), \quad \text{IQR}(y) = |a| \cdot \text{IQR}(x)
\text{SD}(y) = |a| \cdot \text{SD}(x), \quad \text{Var}(y) = a^2 \cdot \text{Var}(x)
Note that the added constant \(b\) does not appear in any spread formula, because adding a constant does not affect variability.
注意常数 \(b\) 不出现在任何离散程度公式中，因为加上常数不影响变异性。
When \(a\) is positive, the transformation is order-preserving. When \(a\) is negative, the data order reverses, but the formulas above still hold for mean, median, and mode, and for the absolute-value scaling of spread.
当 \(a\) 为正数时，变换保持数据顺序。当 \(a\) 为负数时，数据顺序反转，但上述公式对于均值、中位数、众数以及离散程度的绝对值缩放仍然成立。
8. Worked Example with a Small Data Set | 小型数据集的完整示例
Consider the data set \(x\): 1, 3, 5, 7, 9. We apply the transformation \(y = 2x + 1\). First compute the original statistics.
考虑数据集 \(x\)：1, 3, 5, 7, 9。我们应用变换 \(y = 2x + 1\)。首先计算原始统计数据。
Original mean: \(\bar{x} = (1+3+5+7+9)/5 = 5\). Original median: 5. Original range: \(9-1 = 8\). Original IQR: \(Q_3 = 7, Q_1 = 3\), so IQR = 4. Original standard deviation: \(s_x = \sqrt{((1-5)^2 + (3-5)^2 + (5-5)^2 + (7-5)^2 + (9-5)^2)/(5-1)} = \sqrt{(16+4+0+4+16)/4} = \sqrt{40/4} = \sqrt{10} \approx 3.162\).
原始均值：\(\bar{x} = (1+3+5+7+9)/5 = 5\)。原始中位数：5。原始极差：\(9-1 = 8\)。原始四分位距：\(Q_3 = 7, Q_1 = 3\)，所以 IQR = 4。原始标准差：\(s_x = \sqrt{((1-5)^2 + (3-5)^2 + (5-5)^2 + (7-5)^2 + (9-5)^2)/(5-1)} = \sqrt{(16+4+0+4+16)/4} = \sqrt{40/4} = \sqrt{10} \approx 3.162\)。
Using the formulas for \(y = 2x + 1\): \(\bar{y} = 2(5)+1 = 11\); median(y) = 2(5)+1 = 11; range(y) = |2|(8) = 16; IQR(y) = |2|(4) = 8; SD(y) = |2| \times \sqrt{10} = 2\sqrt{10} \approx 6.325; Var(y) = 4 \times 10 = 40.
使用 \(y = 2x + 1\) 的公式：\(\bar{y} = 2(5)+1 = 11\)；中位数(y) = 2(5)+1 = 11；极差(y) = |2|(8) = 16；IQR(y) = |2|(4) = 8；标准差(y) = |2| \times \sqrt{10} = 2\sqrt{10} \approx 6.325；方差(y) = 4 \times 10 = 40。
Let us verify by transforming the data directly: \(y\) values are 3, 7, 11, 15, 19. Mean = (3+7+11+15+19)/5 = 11. Median = 11. Range = 19 − 3 = 16. IQR: Q3 = 15, Q1 = 7, IQR = 8. Standard deviation: deviations from 11 are −8, −4, 0, 4, 8; squared: 64, 16, 0, 16, 64; sum = 160; divide by 4 gives 40; SD = √40 = 2√10. All matches perfectly.
我们通过直接变换数据来验证：\(y\) 值为 3, 7, 11, 15, 19。均值 = (3+7+11+15+19)/5 = 11。中位数 = 11。极差 = 19 − 3 = 16。四分位距：Q3 = 15, Q1 = 7，IQR = 8。标准差：与11的偏差为 −8, −4, 0, 4, 8；平方为 64, 16, 0, 16, 64；总和为 160；除以4得 40；标准差 = √40 = 2√10。所有结果完全一致。
9. Summary Table of Transformation Rules | 变换规则汇总表
| Statistic | Effect of y = x + b | Effect of y = ax | Effect of y = ax + b |
| Mean \bar{x} | \bar{x} + b | a\bar{x} | a\bar{x} + b |
| Median | median + b | a × median | a × median + b |
| Mode | mode + b | a × mode | a × mode + b |
| Range | range | |a| × range | |a| × range |
| IQR | IQR | |a| × IQR | |a| × IQR |
| Standard deviation | SD | |a| × SD | |a| × SD |
| Variance | Var | a² × Var | a² × Var |
In the table, “SD” and “Var” represent the original standard deviation and variance, and “range” and “IQR” represent the original values of those measures.
表中“SD”和“Var”代表原始标准差和方差，“range”和“IQR”代表原始极差和四分位距值。
10. Effects on the Shape of the Distribution | 对分布形状的影响
Adding a constant \(b\) shifts the entire distribution left or right. The shape — including skewness and kurtosis — remains exactly the same. Therefore, any measure based solely on the shape, such as Pearson’s coefficient of skewness, is unchanged when only a constant is added.
加上常数 \(b\) 会将整个分布向左或向右平移。分布的形状——包括偏度和峰度——完全保持不变。因此，任何仅基于形状的度量，如皮尔逊偏度系数，在仅加上常数时都保持不变。
Multiplying by a positive constant changes the scale of the distribution but preserves its shape. The distribution is stretched or compressed horizontally, so relative position measures such as percentiles are still meaningful.
乘以正数常数会改变分布的比例，但保持其形状。分布会被水平拉伸或压缩，因此百分位数等相对位置度量仍然有意义。
Multiplying by a negative constant reflects the distribution about the vertical axis, reversing its skewness. A right-skewed distribution becomes left-skewed, and vice versa.
乘以负数常数会将分布关于纵轴镜像翻转，反转其偏度方向。右偏分布变为左偏分布，反之亦然。
For IB exams, you may be asked how a transformation affects the shape. Remember that adding a constant never changes shape, while multiplying by a constant changes spread but not the general “shape” unless the constant is negative.
在IB考试中，可能会问你变换如何影响形状。记住，加上常数绝不改变形状，而乘以常数会改变离散程度，但除非常数为负，否则不会改变总体“形状”。
11. Common Mistakes and Exam Tips | 常见错误与考试提示
One common mistake is forgetting to use the absolute value when multiplying the range, IQR, or standard deviation by \(a\). For example, if \(a = 3\), the range multiplier is 3, not −3. If \(a = −3\), the multiplier is still 3.
一个常见错误是忘记在将极差、四分位距或标准差乘以 \(a\) 时使用绝对值。例如，如果 \(a = 3\)，极差乘数为3，而不是−3。如果 \(a = −3\)，乘数仍然是3。
Another mistake is applying the shift \(b\) to the variance or standard deviation. Since \(b\) is only added to every data point, it cannot change any spread measure. The final spread depends only on \(a\), not on \(b\).
另一个错误是将平移量 \(b\) 应用于方差或标准差。因为 \(b\) 只是加到每个数据点上，它不可能改变任何离散程度度量。最终的离散程度只取决于 \(a\)，而取决于 \(b\)。
When dealing with grouped data, apply the same formulas. If class midpoints are \(x\) and frequencies remain unchanged, then transforming the midpoints by \(y = ax + b\) transforms the grouped mean and standard deviation using the same rules.
在处理分组数据时，应用相同的公式。如果组中值为 \(x\) 且频数不变，那么将组中值按 \(y = ax + b\) 变换后，分组均值和标准差也遵循相同的规则。
Always check whether the data are a population or a sample. The transformation rules for standard deviation and variance are identical for both population and sample versions, because the divisor \(n\) or \(n-1\) is not affected by a linear transformation of the data.
始终检查数据是总体还是样本。标准差和方差的变换规则对总体和样本版本完全相同，因为除数 \(n\) 或 \(n-1\) 不会受数据的线性变换影响。
12. Real-World Applications | 实际应用
Temperature conversion is a classic example. If \(x\) is temperature in Celsius and \(y\) is temperature in Fahrenheit, then \(y = 1.8x + 32\). The mean temperature in °F is 1.8 times the mean in °C plus 32. The standard deviation in °F is 1.8 times the standard deviation in °C.
温度换算是经典例子。如果 \(x\) 为摄氏温度，\(y\) 为华氏温度，那么 \(y = 1.8x + 32\)。华氏温度均值等于摄氏均值乘以1.8再加32。华氏温度标准差等于摄氏标准差乘以1.8。
Currency conversion works similarly. If an exchange rate is constant, multiplying all prices by the exchange rate changes the mean and standard deviation by the same factor, while adding a fixed transaction fee shifts the mean but not the standard deviation.
货币兑换也类似。如果汇率恒定，将所有价格乘以汇率会使均值和标准差按同一因子变化，而加入固定交易费只会改变均值，不改变标准差。
In physics, converting units from metres to centimetres multiplies all measurements by 100. Thus the mean and standard deviation are also multiplied by 100, while the variance is multiplied by 10,000. This demonstrates why you must square the factor for variance.
在物理中，将单位从米转换为厘米时，所有测量值乘以100。因此，均值和标准差也乘以100，而方差乘以10000。这说明为什么方差必须乘以因子的平方。
These applications show that the transformation rules are not just abstract formulas, but powerful tools for converting between units and interpreting data in new scales.
这些应用表明，变换规则不仅仅是抽象的公式，更是在不同单位之间转换以及在新尺度中解释数据的有力工具。
Conclusion | 结论
In summary, adding a constant \(b\) changes only measures of location: mean, median, and mode all increase by \(b\). Multiplying by a constant \(a\) changes location measures by a factor \(a\) and changes spread measures by factor \(|a|\), with variance changing by \(a^2\).
总而言之，加上常数 \(b\) 只改变位置度量：均值、中位数和众数都增加 \(b\)。乘以常数 \(a\) 会使位置度量乘以因子 \(a\)，使离散程度度量乘以因子 \(|a|\)，而方差乘以 \(a^2\)。
For the combined transformation \(y = ax + b\), use \(\bar{y} = a\bar{x} + b\), \(\text{median}(y) = a\cdot\text{median}(x) + b\), \(\text{SD}(y) = |a|\cdot\text{SD}(x)\), and \(\text{Var}(y) = a^2\cdot\text{Var}(x)\).
对于组合变换 \(y = ax + b\)，使用 \(\bar{y} = a\bar{x} + b\)，\(\text{median}(y) = a\cdot\text{median}(x) + b\)，\(\text{SD}(y) = |a|\cdot\text{SD}(x)\)，以及 \(\text{Var}(y) = a^2\cdot\text{Var}(x)\)。
Mastering these rules will help you solve IB statistics problems quickly and accurately, and will deepen your understanding of how linear transformations affect data.
掌握这些规则将帮助你在IB统计问题中快速准确地求解，并加深你对线性变换如何影响数据的理解。
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