📚 Year 7 CCEA Statistics: High-Frequency Topics and Common Mistakes Analysis | Year 7 CCEA 统计：高频考点与易错题分析
Statistics in Year 7 lays the foundation for all future data handling work. This article highlights the topics that appear most often in CCEA assessments and explains the mistakes students make when tackling them. Reading through each point carefully will help you spot patterns, avoid silly errors, and feel more confident about your class tests and end-of-year exams.
七年级的统计学习为今后所有的数据处理打下基础。本文聚焦 CCEA 考试中反复出现的高频考点，并详细分析学生在解题时的常见错误。认真阅读每一个要点，可以帮助你发现命题规律，避开无谓失分，在课堂测验和期末考试中更自信。
1. Calculating the Mean Accurately | 准确计算平均数
The mean is one of the most frequently tested statistics concepts. Students must add all the values and then divide by the number of values. A typical mistake is forgetting to divide by the correct count, especially when the same number appears more than once. For example, given the set 4, 4, 6, 8, 8, 8, the sum is 38 and there are 6 numbers, so the mean is 38 ÷ 6. Many pupils mistakenly treat the set as 4, 6, 8 and calculate 18 ÷ 3 because they ignore the repeats. Always count every single data point.
平均数是统计中最常考查的概念之一。学生需要将所有数值相加，再除以数值的总个数。一个典型错误是忘记除以正确的个数，尤其是当某个数值重复出现时。例如，给定数据 4, 4, 6, 8, 8, 8，总和为 38，共有 6 个数字，平均数应为 38 ÷ 6。许多学生会误以为数据只有 4, 6, 8 三个，计算 18 ÷ 3，因为他们忽略了重复项。一定要数清每一个数据点。
Another frequent error occurs when a zero is present. If a student records a score of zero, it must be included in both the total and the count. Leaving it out inflates the mean incorrectly. Suppose five test scores are 7, 8, 0, 9, 6. The sum is 30 and the count is 5, so the mean is 6. Omitting the zero gives a sum of 30 and a count of 4, producing a mean of 7.5, which is wrong. Always check your list for zeros before you begin.
另一个常见错误发生在数据中包含零时。如果学生记录了一个零分，必须将零同时计入总和和个数。去掉零会错误地抬高平均数。假设五次测验成绩为 7, 8, 0, 9, 6，总和为 30，数据个数为 5，平均数应为 6。省略零后总和仍为 30，但个数变为 4，得出平均数 7.5，这是错误的。开始计算前，一定要检查列表中是否有零。
When working with larger data sets, using a frequency table helps avoid miscounting. Multiply each value by its frequency, sum those products, then divide by the total frequency. For instance, if the value 5 appears 3 times, the contribution is 5 × 3 = 15, not just 5. Many learners add the values but forget to multiply by the frequency, which drastically reduces the total and the mean. Practise setting out your work in neat columns to minimise errors.
处理较大的数据集时，使用频率表有助于避免计数错误。用每个数值乘以它的频数，将所有乘积相加，再除以总频数。例如，数值 5 出现了 3 次，它的贡献是 5 × 3 = 15，而不仅仅是 5。许多学生虽然加了数值，却忘记乘以频数，这会大大降低总和与平均数。练习用清晰的分栏列出计算过程，可以最大程度减少失误。
2. Median and the Importance of Ordering | 中位数与排序的重要性
The median is the middle value when data is arranged in order. The single biggest mistake Year 7 students make is failing to put the numbers in ascending or descending order first. If you try to pick the middle from an unordered list, you are almost certain to get the wrong answer. For example, with the raw data 11, 5, 8, 3, 10, many pupils will quickly say the middle is 8 because it sits in the centre of the list as written. The correct ordered list is 3, 5, 8, 10, 11, giving a median of 8 here by coincidence, but when the numbers are 11, 15, 8, 3, 10 the unordered middle is 8 and the true median is 10. Always order first.
中位数是将数据按顺序排列后处于中间位置的数值。七年级学生最大的错误是没有先把数字按从小到大或从大到小排序。如果直接从无序列表中取中间值，几乎肯定会答错。例如，面对原始数据 11, 5, 8, 3, 10，很多学生快速说出中间是 8，因为它在书写列表的正中央。正确的排序是 3, 5, 8, 10, 11，这里中位数恰好也是 8，但若数据为 11, 15, 8, 3, 10，无序中间值是 8，而真正的中位数是 10。一定要先排序。
When the data set has an even number of values, the median is the mean of the two middle numbers. Pupils often forget this step and simply pick one of the middle numbers, or they become confused about which two numbers to choose. Take the set 4, 7, 9, 12. The two middle numbers are 7 and 9, so the median is (7 + 9) ÷ 2 = 8. A common error is stating the median is 7 or 9, or even trying to find a single number in the middle that does not exist. Highlight the two central positions after ordering, and then work out their average.
当数据个数为偶数时，中位数是中间两个数的平均数。学生常常忘记这一步，只简单选取中间两数中的一个，或者搞不清该选哪两个数。以 4, 7, 9, 12 为例，中间两个数是 7 和 9，因此中位数为 (7 + 9) ÷ 2 = 8。常见错误是说中位数是 7 或 9，甚至试图找出一个并不存在的中间数。排序后标出中间两个位置，再计算它们的平均数。
Another trap is misreading the question when it asks for the median from a frequency table. You need to use the total frequency to locate the middle position, then work through the table to find which value occupies that position. Suppose a table shows the number of books read: 0 books (frequency 2), 1 book (frequency 5), 2 books (frequency 3). The total frequency is 10, so the median lies between the 5th and 6th data points. Both the 5th and 6th points fall in the ‘1 book’ category, so the median is 1. Rushing to the middle of the value column is a frequent mistake.
另一个陷阱是当题目要求从频率表中找出中位数时读错题意。你需要利用总频数确定中间位置，再顺着表格找到该位置对应的数值。假设表格显示阅读书本数：0 本（频数 2），1 本（频数 5），2 本（频数 3）。总频数为 10，因此中位数位于第 5 和第 6 个数据点之间。第 5 和第 6 个点都落在“1 本”这一组，所以中位数是 1。直接跳到数值列的中间是常见错误。
3. Mode Misunderstandings | 众数理解的误区
The mode is the value that appears most often. Many pupils lose marks by confusing the mode with the highest value in a data set. If the numbers are 3, 7, 7, 2, 9, the mode is 7 because it appears twice, whereas the highest value is 9. Stating the mode as 9 is a very common slip. Always ask yourself: ‘Which number occurs most frequently?’ not ‘Which number is largest?’
众数是出现次数最多的数值。许多学生因为混淆众数与数据中的最大值而丢分。如果数字为 3, 7, 7, 2, 9，众数是 7，因为它出现了两次，而最大值是 9。将众数填成 9 是非常常见的疏忽。始终问自己：“哪一个数字出现得最频繁？”而不是“哪一个数字最大？”
Another misconception is that every data set must have exactly one mode. Some sets may have two modes (bimodal) or no mode if all values occur with the same frequency. For instance, the set 4, 5, 6, 7 has no mode because every number appears once. Writing 4, 5, 6, and 7 as four modes is acceptable in some contexts, but CCEA mark schemes at this level usually expect ‘no mode’ or simply stating that all values appear once. Check with your teacher about the specific wording, but be aware that a mode is not guaranteed.
另一个误解是认为每个数据集都必须恰好有一个众数。有些数据集可能有两个众数（双众数），或者如果所有数值出现频率相同则没有众数。例如，4, 5, 6, 7 这个数据集没有众数，因为每个数字都只出现一次。将 4, 5, 6, 7 都列为众数在某些情况下可以接受，但这一阶段的 CCEA 评分标准通常要求回答“无众数”或写明所有数值都只出现一次。具体措辞请与老师确认，但要意识到众数并非必然存在。
In grouped frequency tables, the modal class is the class interval with the highest frequency, not the class with the largest midpoint. A classic error is to look at the intervals and pick the one with the biggest numbers. For example, if the intervals 0–4, 5–9, 10–14 have frequencies 3, 8, 2 respectively, the modal class is 5–9. Do not say the mode is 10–14 just because it contains the largest numbers. The frequency determines the mode.
在分组频率表中，众数所在的组是频数最高的那一组，而不是中点最大的一组。一个经典错误是看着区间，挑选包含最大数字的那一组。例如，区间 0–4、5–9、10–14 的频数分别为 3、8、2，则众数所在的组是 5–9。不要因为 10–14 含有最大数字就说它是众数。频数决定众数。
4. Range Calculation Errors | 极差计算错误
The range is found by subtracting the smallest value from the largest value. It measures the spread of the data. The most common mistake is a simple subtraction error, especially when negative numbers are involved or when the data set is large. Take 12, 3, 19, 7, 5. The largest is 19, the smallest is 3, so the range is 19 – 3 = 16. Sometimes pupils reverse the order and subtract largest from smallest, which would give a negative range; a negative range is a clear signal that a mistake has been made.
极差是用最大值减去最小值得到的。它衡量数据的散布程度。最常见的错误是简单的减法失误，尤其是在数据包含负数或数据集较大时。以 12, 3, 19, 7, 5 为例，最大值为 19，最小值为 3，极差为 19 – 3 = 16。有时学生会弄反顺序，用最小值减去最大值，得出负数极差；负数极差是出现错误的明显信号。
Another slip occurs when pupils identify the largest and smallest numbers from a frequency table but fail to use the actual values. In a table showing scores and their frequencies, the largest score is the highest value in the left-hand column, and the smallest is the lowest value—provided those scores actually appear. A student might incorrectly use the frequency numbers instead. For instance, if the table shows heights 130 cm (freq 4), 135 cm (freq 2), 140 cm (freq 5), the range involves the heights, not the frequencies. The range is 140 – 130 = 10 cm, not 5 – 2 = 3. Always read the table headings carefully.
另一个疏忽是学生从频率表中圈定最大、最小值时，没用对实际数值。在一个显示分数及其频数的表格中，最大值是左栏的最高数值，最小值是最低数值——前提是这些分数确实出现。学生可能会错误地使用频数来求极差。例如，表格显示身高 130 cm（频数 4）、135 cm（频数 2）、140 cm（频数 5），极差涉及的是身高，而非频数。极差为 140 – 130 = 10 cm，而不是 5 – 2 = 3。一定要仔细阅读表头。
When the data is grouped, the range can only be estimated by subtracting the lower boundary of the first class from the upper boundary of the last class. CCEA Year 7 questions usually avoid this complexity, but you might be asked to find the range from a set of individual data points even when a grouped table is shown. Stay alert: if the question asks for the range of the original data, use the raw numbers if given.
当数据被分组后，极差只能通过用最后一组的上界减去第一组的下界来估算。CCEA 七年级题目通常会避开这种复杂性，但即使给出了分组表格，你也可能被要求根据给出的个别数据点求极差。保持警惕：如果问题要求原始数据的极差，只要给出了具体数字，就要使用它们。
5. Bar Charts vs. Histograms: Avoiding Confusion | 条形图与直方图的混淆
At Year 7 level, CCEA focuses on bar charts for discrete or categorical data. A bar chart has gaps between the bars to show that the categories are separate. Many students draw bars that touch, turning the chart into a histogram. This loses marks under presentation criteria. Always leave equal, consistent gaps between bars unless the question specifically asks for a histogram (which is rare at this stage).
在七年级阶段，CCEA 重点关注用于离散或分类数据的条形图。条形图的条形之间有间隙，以表示类别各自独立。许多学生会画出紧挨着的条形，把图变成了直方图。这会按呈现标准被扣分。除非题目明确要求画直方图（此阶段很少见），否则务必在条形之间留出均匀、一致的间隙。
Labeling axes is another high-frequency assessment point. The horizontal axis should name the categories, and the vertical axis should show the frequency and be clearly numbered. A common mistake is forgetting to write what the axes represent, or using a scale that is not linear. For example, if the vertical axis jumps from 0 to 2, then 4, then 10, the chart is misleading. Use a proper scale with even steps: 0, 2, 4, 6, 8, etc. Also, always write the frequency numbers on the lines, not in the spaces between them.
轴标签是另一个高频评分点。横轴应标注类别名称，纵轴应显示频数并清晰标上数字。常见错误是忘记写明轴所代表的含义，或者使用了不均匀的比例尺。例如，纵轴从 0 跳到 2，再到 4，接着跳到 10，这样的图表会误导读者。应使用步长均匀的合适比例尺：0, 2, 4, 6, 8 等。此外，务必将频数标注在刻度线上，而不是线之间的空白处。
Pupils often misread bar chart questions that ask ‘How many more…’ or ‘What is the difference…’. They may simply state the frequency of the larger bar instead of subtracting the two frequencies. If a bar for cats reaches 8 and a bar for dogs reaches 5, then ‘how many more cats than dogs?’ requires the calculation 8 – 5 = 3. Writing 8 is a typical error. Underline the comparison word in the question to remind yourself to subtract.
学生经常误读条形图题目中的“多多少……”或“……的差是多少”。他们可能只写出较高条形对应的频数，而没有减去另一个频数。如果代表猫的条形读数为 8，代表狗的读数为 5，那么“猫比狗多多少？”需要计算 8 – 5 = 3。写成 8 是典型错误。在题目中划出比较性的词语，提醒自己要做减法。
6. Pie Chart Angle Calculations | 饼图的角度计算
Pie charts are introduced in Year 7, and calculating the angle for each sector causes difficulty. The total angle in a pie chart is 360°, and each category’s angle is found by (category frequency ÷ total frequency) × 360. A widespread error is dividing by the category frequency instead of multiplying, or forgetting to multiply by 360 altogether. For instance, if 10 out of 40 students walk to school, the fraction is 10/40 = 1/4, so the angle is 1/4 × 360 = 90°. Many pupils stop at 1/4 and do not convert it to degrees.
饼图在七年级被引入，计算每个扇形的角度是一大难点。饼图的总角度为 360°，每个类别的角度通过（类别频数 ÷ 总频数）× 360 来求得。一个普遍错误是用类别频数去除其他数，而不是乘；或者完全忘记乘以 360。例如，40 名学生中有 10 名步行上学，比例是 10/40 = 1/4，因此角度为 1/4 × 360 = 90°。许多学生算到 1/4 就停下了，没有将其转换为角度。
Measuring and drawing the angles with a protractor also presents practical challenges. Common mistakes include reading the wrong scale on the protractor (inside vs. outside), not aligning the centre point with the vertex of the angle, or drawing the lines too short to measure accurately. When constructing a pie chart in an exam, use a sharp pencil, draw a clear radius to start, and mark small ticks at the correct degree before ruling the line. Double-check that the sum of your sector angles is close to 360°; a sum of 358° or 362° indicates a minor drawing or rounding error.
使用量角器测量和绘制角度也存在实际操作上的困难。常见错误包括看错量角器的刻度（内圈与外圈混淆）、没有将中心点与角的顶点对齐，或者画的线条太短导致无法准确测量。在考试中绘制饼图时，使用削尖的铅笔，先画出一条清晰的起始半径，在正确角度处做小标记，再用尺子连线。检查所有扇形的角度之和是否接近 360°；总和为 358° 或 362° 说明有轻微的画图或取整误差。
A reasoning question might ask: ‘Is it possible to draw a pie chart if the total frequency is missing?’ The answer is no, because you need the total to work out each proportion. Some pupils try to draw a pie chart directly from given percentages. While percentages can be converted to angles by multiplying by 3.6 (since 1% = 3.6°), they must first verify that the percentages add up to 100%. A pie chart with percentages summing to 90% or 110% is invalid.
可能的推理题会问：“如果总频数缺失，能否绘制饼图？”答案是不能，因为你需要总频数来计算每个部分的比例。有些学生试图直接根据给定的百分比来画饼图。虽然百分比可以乘以 3.6（因为 1% = 3.6°）转换为角度，但他们必须先确认百分比加起来等于 100%。百分比总和为 90% 或 110% 的饼图是无效的。
7. Data Collection and Survey Bias | 数据收集与调查偏差
CCEA frequently asks Year 7 students to critique survey methods or suggest improvements. A leading question is one that encourages a particular answer, for example, ‘Don’t you agree that homework is boring?’ Biased questions can make data unreliable. A better version would be: ‘How do you feel about the amount of homework you receive?’ with options ranging from ‘Far too little’ to ‘Far too much’.
CCEA 经常要求七年级学生评价调查方法或提出改进建议。诱导性问题是指那些鼓励特定回答的问题，例如：“你不觉得家庭作业很无聊吗？”带有偏见的问题会使数据不可靠。更好的版本是：“你对目前布置的家庭作业量有何感受？”并提供从“太少”到“太多”的一系列选项。
Sample size and selection are another major topic. Pupils often suggest surveying ‘everyone in the country’ for a school project, which is not practical. A sample needs to be large enough to be representative but small enough to be manageable. A common exam pitfall is identifying a sample that is too small or biased. For example, asking only the school football team about sports facilities will not represent the whole school. Always consider who is included and who is left out.
样本大小和选择方式是另一个重要话题。学生常常建议为了一个学校项目去调查“全国的每一个人”，这不切实际。样本需要足够大以具有代表性，但又不能大到难以操作。考试中常见的陷阱是认定一个样本太小或存在偏差。例如，只问学校足球队关于体育设施的看法无法代表全校。要始终考虑谁被包含进来，谁被排除在外。
A typical question might present a scenario where a student stands at the school gate at 8:30 a.m. and asks every 10th person entering. This is a reasonable systematic sample because it spreads across different forms and staff. But if the student only asks their friends in one year group, that is a convenience sample and likely biased. Be ready to explain why a sampling method is fair or unfair using the words ‘representative’ and ‘bias’.
典型的题目可能会描述这样一个场景：一名学生早上 8:30 站在校门口，每隔 10 个人询问一位进入者。这是一个合理的系统抽样，因为它涵盖了不同的班级和教职工。但如果这名学生只问自己同一年级的朋友，那就是便利抽样，很可能存在偏差。准备好用“代表性”和“偏差”这两个词解释某种抽样方法的公正与否。
Recording data with tally marks and frequency tables is a skill that seems simple but often leads to mistakes under time pressure. The most common slip is forgetting to cross the fifth tally to make a gate of five. This makes counting the total far slower and more error-prone. Always draw tallies in groups of five: four vertical marks and a diagonal slash through them. Then write the numerical frequency clearly next to the tallies.
用画记法和频率表记录数据是一项看似简单，但时间压力下常出错的技能。最常见的疏忽是忘记将第五个画记横穿过去组成“正”字的五画。这使得计数总频数变得更慢且更容易出错。始终以五个为一组画记：四个竖画加一条斜线穿过。然后在画记旁边清晰地写出数字频数。
8. Reading and Interpreting Tables and Charts | 表格与统计图表的解读
Many marks in CCEA statistics assessments are allocated to interpreting information from two-way tables, pictograms, and line graphs. With two-way tables, pupils often read from the wrong row or column. For example, a table showing favourite subjects by gender: the cell intersecting ‘Boys’ and ‘Art’ gives the number of boys who prefer Art. Always trace your finger across and down to confirm the intersection before writing your answer.
CCEA 统计评估中的很多分值用于考查对双向表、象形图和折线图的解读能力。面对双向表，学生常常读错行或列。例如，一张按性别划分最喜爱科目的表格：“男生”行与“艺术”列交叉的单元格给出喜欢艺术的男生人数。答题前，始终用手指横向、纵向移动，确认交叉点。
Pictograms use symbols to represent a certain number of items. The key is essential; a common error is assuming one symbol equals one unit when the key states one symbol represents 2 or 5 units. If a pictogram shows 3 and a half apples and each apple stands for 4 children, then the frequency is 3.5 × 4 = 14. Some students simply count the symbols as 3 or 4 without multiplying. Always read the key first and underline it.
象形图用符号表示特定数量的项目。图例至关重要；常见的错误是以为一个符号代表一件物品，但图例却注明一个符号代表 2 或 5 个单位。如果一个象形图显示了 3 个半苹果，且每个苹果代表 4 名儿童，那么频数为 3.5 × 4 = 14。有些学生仅仅将符号数成 3 或 4，却不进行乘法运算。始终先读图例并划线标出。
Line graphs are frequently used to show changes over time. The most frequent blunder is misreading the scale on the time axis or assuming a steady trend continues without fail. When a line graph shows temperature over a week, pupils may be asked: ‘Between which two days did the temperature rise the most?’ This requires calculating the difference for each consecutive pair of days, not just looking at the steepest line segment by eye. A segment from 5°C to 11°C is steeper than one from 10°C to 14°C and also records a larger increase, but the visual must be checked with subtraction. Always do the arithmetic.
折线图经常用来展示随时间变化的情况。最常见的错误是读错时间轴的刻度，或假定趋势会一成不变地延续下去。当折线图显示一周内的温度变化时，学生可能被问到：“哪两日之间温度上升最多？”这需要计算每一对连续日子的差值，而不是仅凭肉眼看出最陡的线段。例如，从 5°C 到 11°C 的线段比从 10°C 到 14°C 的更陡，并且上升幅度也更大，但视觉判断必须通过减法来验证。始终进行算术计算。
9. Spotting Patterns and Drawing Conclusions | 发现规律与得出结论
Statistics is not only about calculating numbers; it is about telling a story from the data. CCEA expects Year 7 students to make simple comparisons and state what the data shows. A weak answer says ‘The bar chart is about pets.’ A stronger answer says ‘The bar chart shows that cats are the most popular pet in Year 7, chosen by 15 students, while only 2 students have fish.’ Always refer to the figures and categories specifically.
统计不仅仅是计算数字，更是通过数据讲述一个故事。CCEA 期待七年级学生能够进行简单的比较，并说明数据表明了何种趋势。较弱的回答是：“这个条形图是关于宠物的。”较强的回答是：“条形图显示，猫是七年级最受欢迎的宠物，有 15 名学生选择，而只有 2 名学生养鱼。”务必具体地引用数据和类别。
When comparing two data sets, be systematic. State the average for each set (mean, median, or mode whichever is appropriate) and then compare the spreads using the range. For instance, ‘Class A has a mean score of 72% with a range of 40%, while Class B has a mean score of 70% with a range of 20%. This suggests Class A has a slightly higher average but more variation in scores.’ Avoid vague phrases like ‘Class A did better’. Provide evidence.
对比两组数据时，要有系统性。先陈述每组数据的平均数（平均数、中位数或众数，视情况而定），然后用极差比较其分散程度。例如，“A 班的平均分为 72%，极差为 40%，而 B 班的平均分为 70%，极差为 20%。这表明 A 班的平均水平略高，但分数差异较大。”避免使用“A 班表现更好”这样模糊的表述。要提供证据。
A frequent mistake is drawing conclusions that go beyond the data. If a survey of 30 Year 7 students shows that 20 prefer pizza, you cannot conclude ‘All children love pizza.’ The correct conclusion is ‘The majority of the students surveyed prefer pizza.’ CCEA mark schemes penalise over-generalisations. Keep your conclusion tightly linked to the sample described in the question.
一个常见错误是得出超越数据范围的结论。如果对 30 名七年级学生的调查显示 20 人喜欢披萨，你不能得出“所有孩子都爱吃披萨”的结论。正确的结论是：“在受调查的学生中，大多数人更喜欢披萨。”CCEA 的评分标准会对过度泛化进行扣分。确保你的结论紧扣题目中描述的样本。
10. Common Arithmetic Slips in Statistics Problems | 统计题中的常见算术失误
Under exam pressure, simple arithmetic errors creep into statistical calculations. Adding a column of numbers incorrectly is a leading cause of lost marks. To prevent this, always add the numbers twice—once down the column and once up—to see if you get the same sum. Alternatively, group numbers that make ten or a multiple of ten before adding the rest. For example, with 7, 3, 8, 2, 5, group (7+3) and (8+2) for 10+10+5 = 25.
在考试压力下，简单的算术错误会悄然出现在统计计算中。将一列数字错误相加是失分的主要原因。为避免这种情况，务必将数字加两遍——一次从上往下加，一次从下往上加——看两次的总和是否相同。或者，先将能凑成十或十的倍数的数字进行组合，再添加剩余的数字。例如，对于 7, 3, 8, 2, 5，可将 (7+3) 和 (8+2) 组合得到 10+10+5 = 25。
Division errors when finding the mean are also widespread, especially with long division. A pupil might correctly sum five numbers as 147 but then divide by 4 instead of 5, or make a mistake in the division itself. A quick check is to multiply your mean by the number of values; it should return the original total (or very close to it if rounding). For 147 ÷ 5 = 29.4, checking 29.4 × 5 gives 147. If you get 36.75 × 4 = 147, you divided by 4 mistakenly. Use estimation to see if your answer is sensible.
求平均数时的除法错误也很常见，尤其是做长除法。学生可能正确地将五个数字相加得到 147，但在除以时误用了 4 而不是 5，或在计算除法本身时出了错。一个快速检查的方法是：将你的平均数乘以数据个数，应该得到原本的总和（如果涉及四舍五入，结果应非常接近）。对 147 ÷ 5 = 29.4，检查 29.4 × 5 得到 147。如果你得到 36.75 × 4 = 147，说明你错误地除以了 4。用估算来判断你的答案是否合理。
With percentages and proportions, pupils often forget to multiply by 100 when converting a fraction to a percentage. For instance, if 17 out of 25 students bring a packed lunch, the proportion is 17/25 = 0.68. To write this as a percentage, multiply by 100: 0.68 × 100 = 68%. Many stop at 0.68 and think the answer is 0.68%. Understanding that percentages are out of 100 is fundamental. Always ask: ‘Is my answer out of 100?’
在处理百分比和比例时，学生经常忘记将分数转换为百分比时要乘以 100。例如，25 名学生中有 17 名自带午餐，比例为 17/25 = 0.68。要写成百分比形式，需乘以 100：0.68 × 100 = 68%。许多学生算到 0.68 就停住了，以为答案是 0.68%。理解百分比是以 100 为基数是根本所在。始终问自己：“我的答案是基于 100 的吗？”
A final common error is misplacing the decimal point when using calculators. A mean of 4.5 can mistakenly become 45, or a percentage of 32% can become 3.2% if the decimal is shifted. Always do a rough mental estimate first. If the numbers are mostly between 0 and 10, a mean of 45 is impossible. Sanity-checking your answers catches many calculator mistakes before you write them down.
最后一种常见错误是在使用计算器时点错了小数点。原本 4.5 的平均数可能误变成 45，或者 32% 的百分比因为小数点移位变成 3.2%。始终先进行粗略的心算估计。如果数据大部分介于 0 到 10 之间，平均数为 45 是不可能的。在下笔前对答案进行合理性检验，能在书写前发现许多计算器错误。
Published by TutorHao | Statistics Revision Series | aleveler.com
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