Year 9 CAIE Statistics: Formula & Theorem Quick Reference Handbook | Year 9 CAIE 统计：公式定理速查手册

📚 Year 9 CAIE Statistics: Formula & Theorem Quick Reference Handbook | Year 9 CAIE 统计：公式定理速查手册

Welcome to your essential revision companion for Year 9 CAIE Statistics. This handbook provides a concise summary of all the formulas, definitions and key theorems you need to master. Whether you are calculating averages, constructing charts or solving probability problems, keep this guide close at hand for quick reference.

欢迎使用这份 Year 9 CAIE 统计学必备复习指南。本手册简明扼要地汇总了所有需要掌握的公式、定义与关键定理。无论你在计算平均数、绘制图表还是解答概率问题，都可以随时参考这本指南。

1. Measures of Central Tendency: Mean, Median, Mode | 集中趋势测量：平均数、中位数与众数

The mean is the most commonly used measure of central tendency. It is found by adding together all the values in a data set and then dividing by the number of values.

平均数是衡量集中趋势最常用的量数。将数据集中所有数值相加，再除以数据的个数，即可得到平均数。

x̄ = ∑x / n

In this formula, ∑x represents the sum of all data items, and n is the total frequency (count) of the data set.

公式中，∑x 表示所有数据项的总和，n 为数据集的总频数（即数据个数）。

The median is the middle value when the data are arranged in order. If there is an odd number of observations, the median is the value at position (n+1)/2. If there is an even number, the median is the average of the two middle values (positions n/2 and n/2 + 1).

中位数是将数据按大小顺序排列后处于中间位置的数值。若观测值个数为奇数，中位数是第 (n+1)/2 个位置上的值；若为偶数，中位数是中间两个值（第 n/2 和第 n/2+1 个位置）的平均数。

The mode is the value that occurs most frequently in a data set. A data set may have one mode, more than one mode (bimodal or multimodal) or no mode if all values occur with equal frequency.

众数是在数据集中出现次数最多的值。一组数据可能只有一个众数，也可能有多个众数（双众数或多众数）；如果所有值出现的频率相同，则没有众数。

2. Measures of Spread: Range | 离散程度测量：全距

The range is a simple measure of spread, showing how far apart the smallest and largest values are.

全距是一种简单的离散程度测量方法，反映数据中最小值与最大值之间的差距。

Range = Maximum value – Minimum value

The range is affected by outliers and should be used together with other measures for a full picture of dispersion.

全距容易受极端值影响，应与其他度量指标结合使用，才能全面了解数据的离散情况。

3. Calculating the Mean from a Frequency Table | 从频数表计算平均数

When data are presented in a frequency table, the mean is calculated using the formula involving the frequencies (f) and the data values (x).

当数据以频数表形式呈现时，平均数可通过包含频数 (f) 和数据值 (x) 的公式来计算。

x̄ = ∑(f × x) / ∑f

Here, f is the frequency of each value, and x is the corresponding data value. Multiply each value by its frequency, sum these products, and then divide by the total frequency.

式中，f 为每个值的频数，x 为相应的数据值。将每个值乘以其频数，求出这些乘积的总和，再除以总频数。

Example: A frequency table shows the number of pets per household. Values (x): 0, 1, 2, 3 with frequencies (f): 4, 7, 5, 2. Then ∑(f × x) = (0×4)+(1×7)+(2×5)+(3×2)=0+7+10+6=23, ∑f=18, mean = 23/18 ≈ 1.28.

示例：某频数表显示每个家庭拥有宠物的数量。值 (x)：0, 1, 2, 3；对应频数 (f)：4, 7, 5, 2。则 ∑(f×x) = (0×4)+(1×7)+(2×5)+(3×2)=0+7+10+6=23，∑f=18，平均数 = 23/18 ≈ 1.28。

You can also add an extra column to the frequency table to show the product f × x before summing.

你也可以在频数表中增加一列来展示 f × x 的乘积，然后再求和。

4. Finding the Median from a Frequency Table | 从频数表求中位数

To find the median from a frequency table, first calculate the cumulative frequency. The median position is given by (Total frequency + 1) / 2 if you are using the standard convention for discrete data.

要从频数表求中位数，首先计算累积频数。根据离散数据的常规约定，中位数的位置为 (总频数 + 1) / 2。

Locate the cumulative frequency that first reaches or exceeds the median position. The corresponding data value is the median. If the median position falls exactly between two values, take the average of the two data values.

找到首次达到或超过中位数位置的累积频数，其对应的数据值即为中位数。如果中位数位置恰好落在两个值之间，则取这两个值的平均数。

For grouped

Published by TutorHao | Year 9 统计 Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）