الرياضيات
The Arithmetic Mean
Total divided by count, worked on five values. Also the backwards question — finding a missing value from the mean — and what one extreme value does to it.
Arithmetic mean
The arithmetic mean is the total of a set of values divided by how many values there are.
- يُسمّى أيضاً
- Mean, Average
- أين يقابله الطلاب
- Grade 5, and then in every statistics topic that follows, including the mean from a frequency table at IGCSE and data analysis on the Digital SAT.
الإجابة باختصار
The arithmetic mean is the sum of a set of values divided by the number of values. For 7, 9, 12, 14 and 18 the total is 60 and there are five values, so the mean is 12. It is what people usually mean when they say the average.
مثال
(7 + 9 + 12 + 14 + 18) ÷ 5 = 60 ÷ 5 = 12
Add every value once, then divide by how many there were. The result does not have to be one of the original values and does not have to be a whole number — a mean of 12.4 goals or 2.3 children is perfectly correct, even though no match ends in 12.4 goals. The mean is a summary of the set, not a member of it.
Sum and count, and nothing else
The mean is the number that every value would have if the total were shared out equally. That is worth saying out loud, because it explains why the sum matters and the order does not, and why adding a value equal to the current mean leaves the mean unchanged.
Two mistakes account for most of the lost marks. The first is dividing by the wrong count — from a frequency table you divide by the total frequency, not by the number of rows. The second is forgetting to include a zero: if one of five students scored nothing, the count is still five.
The question that runs backwards
Rearranged, the definition says total = mean × count, and that single line answers a whole family of exam questions. If four test marks are 6, 11, 13 and 19 and a fifth mark is needed to give a mean of 14, the five marks must total 5 × 14 = 70. The four known marks total 49, so the missing one is 21.
The same move handles combining two groups. A class of 20 with mean 62 contributes a total of 1,240; a class of 30 with mean 57 contributes 1,710. Together that is 2,950 across 50 students, so the combined mean is 59 — not 59.5, which is what averaging 62 and 57 would wrongly give, because the larger class pulls harder.
One extreme value moves it a long way
Because the mean uses the size of every value, a single unusual one drags it. The set 2, 3, 4, 5, 96 has a mean of 22, which is larger than four of the five values in it. Describing that data as "averaging 22" is technically true and practically misleading.
That is a limitation to know, not a reason to distrust the mean. The mean is the right summary when the total genuinely matters — total pay, total rainfall, total marks — precisely because it is built from the total. What it does not do is tell you what a typical value looks like when the data has a long tail, and noticing which of those two questions is being asked is the actual skill.
أسئلة شائعة
Is the mean the same as the average?
In everyday speech, yes. In mathematics, "average" is a general word covering the mean, the median and the mode, and "arithmetic mean" is the precise name for the sum-divided-by-count one. Exam questions usually say mean when they want that calculation, and say average only when the choice is part of what is being tested.
How do I find the mean from a frequency table?
Multiply each value by its frequency, add those products to get the overall total, then divide by the total frequency. A common slip is dividing by the number of rows in the table instead. For grouped data you use the midpoint of each class, which makes the answer an estimate rather than an exact mean.
Can the mean be a decimal when the data are whole numbers?
Yes, and it usually is. A mean of 2.4 children per family is a correct summary of whole-number data; it describes the total shared out equally, not an actual family. Rounding it to 2 throws away information the question was asking for, so leave it as a decimal unless told otherwise.
What is a weighted mean?
A mean where some values count for more than others. If coursework is worth 30% and an exam 70%, a student scoring 80 and 60 has a weighted mean of 0.3 × 80 + 0.7 × 60 = 66, not 70. Combining the means of two differently sized groups is the same idea, with group size as the weight.
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