What Are the Mean, the Median, and the Mode?

By Mark Nichol

Among the various statistical varieties of averages, know the distinction between the three most common: the mean, the median, and the mode. Writers and editors don’t necessarily have to be working in the realms of science or mathematics to be confronted with one or more of these terms, so it’s best to be familiar with each one and how it differs from the others.

The mean is the type of average most people are used to. To find the mean for a set of numbers, add the numbers together and divide by the number of numbers in the set. For example, if you roll two dice thirteen times and get 9, 4, 7, 6, 11, 9, 10, 7, 9, 7, 11, 5, and 4, add the numbers to produce a sum of 99. Divide that number by 13 to get 7.6 (rounded off to one decimal point), the mean of that set of numbers.

To find the median, sort the numbers by value (4, 4, 5, and so on); the median is the middle number (here, seventh out of thirteen), or 7. The mode is the number that appears most often — in this case, a tie for 7 and 9 (easier to find once you’ve sorted the numbers).

Another factor is the range, the distance between the smallest and largest numbers. Based on the results for the mean, the median, and the mode of the number set above, you can guess the number before you figure it. You’re right: 11 minus 4 equals 7. Every solution in the four operations I just did came out to 7 or thereabouts — though 9 tied with 7 as the mode).

This shouldn’t be surprising: Because I used the roll of two dice to come up with the numbers in the set, the range was fairly tight (12 minus 2 equals 10), so the chances of hitting the same number for the mean, the median, and the range were high. A larger possible range, and a larger set of numbers, would increase the chances of producing different figures.

For that reason, too, none of the resulting values for the mean, the median, the mode, or the range need to correspond to any of the numbers in the set.

As a side note, accept the incontrovertible fact that no matter how many times you roll two dice, your chances of getting a certain number never change: Just as a coin toss is always a 50-50 proposition, you always have a 1-in-11 chance of getting snake eyes, a 1-in-11 chance of rolling a lucky 7, and so on. But try telling a gambler that.

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16 Responses to “What Are the Mean, the Median, and the Mode?”

  • gS49

    You’re better at English than math.

    There’s a 1 in 36 chance of rolling a 2 on two dice; same chance for a 12. The chance of rolling 7 is 1 in 6.

    There is only one combination that will produce a 2: 1 on one die and 1 on the other. But there are six combinations that produce a 7.

    Sorry, but I have to give you a D in math.

  • Joey Espinosa

    The last paragraph is correct in principle, but wrong in the math. Rolling snake eyes with 2 six-sided dice has a 1/36 probability, sand rolling a seven is 1/6.

  • crashmstr

    I have a headache (so my thinking is not at it’s best), but your closing comment on the probability of getting snake eyes being equal to getting 7 is not right.

    You have one and only one way to roll a 2 – both dice come up 1.

    But with 7, you could get 6+1, 5+2, or 4+3.

    With two or more dice, numbers in the middle get more likely because there are more ways to roll them.

  • Stephen

    You covered the averages really well, but I’m afraid your probability paragraph at the end is way off. There is not an equal chance of getting any result when you roll two dice, because they are rolled independently of each other. There are 36 possible outcomes. There is a 1-in-36 chance of getting a snake eyes (because there is only one way that a snake eyes can happen: both dice show a 1), but there is a 1-in-6 chance of rolling a 7 (because there are 6 different ways of rolling a 7: 1 6, 2 5, 3 4, 4 3, 5 2, 6 1).

    Also, the mode does have to be a member of the set, but a set can contain more than one mode.

  • Cindy Cotter

    I’m happy to see a post about statistics; I’m a big fan of numerical data. For the sake of everyone who thinks of the mode and median as peculiar abstractions, perhaps it would be helpful to practical examples of instances where the median or the mode would be preferable to the mean.

    One quibble: Shouldn’t the title be “What ARE….”?

  • Mark Nichol

    Thanks for the corrections. I obviously didn’t do my math homework (several other visitors responded to me privately about the problems with my probability), but my point was simply to point out the differences between these averaging procedures (which, yes, I should have introduced in the headline with a plural verb).

  • Daniel Scocco

    The mistake in the title was my fault, not Mark’s. Sorry about that.

  • katik aggarwal

    thanks for the statistical class in between. 🙂

  • mrburkemath

    Another way to address the median is that half of the data will be below that amount (50%) and half will be above it.

    This is why the median household income is so much lower than the average (or mean) household income. The mean adds in all those celebrity superstars and CEOs. With the median, they count as one person, just like the guy making minimum wage.

  • ApK

    I wish you folks had stayed quiet on the dice thing, at least until after the crap game I was going to play with Mark.

    Guess I have to find…um…another mark….

  • Mark Nichol

    ApK:

    Perhaps a game of cards instead. I’m terrible at poker, too.

  • Max King

    Varieties of averages ? pedantically, measures of central tendency. Other than in scientific or statistical writings, or a journalist trying to be clever, I could not imagine an author referring to a modal value, or a median (or a geometric mean). Most people would identify the average with the arithmetic mean.

    With increasingly larger samples of measurements from your dice throwing, the values of the mode and median converge towards the mean.

    And, the property that is described in the last paragraph refers to mutually exclusive events – i.e. the outcome of an event is not affected by previous events (or future events, but that is an abstraction too far).

  • Cindy Cotter

    Max, I think people use the mode fairly often without thinking of it as a mode. In fact, they’re apt to make fun of the unwary who use the mean in the wrong place: The average American family has 2.2 children, or The average American homeowner own 1.04 houses, or The most common shoe size for American women is 6.8. (I made all those numbers up.)

    The median is a little trickier, but I think mrburkemath’s example of house prices is on the mark. In some cases, talking about salaries or housing prices, a good reporter really does need to use the median to be clear, not merely clever.

  • Max King

    Cindy, a couple of points.
    Firstly, the mode is the most common occurrence/measurement; I do not know of any Australian families with 2.2 children, let alone 2.2 children being the most common number of children in a family.

    Secondly,in statistical terms the “average XXXXXX family” is a furphy. However it is possible to say ” the average number of children in American families is 2.2″; or “the average number of houses owned by American homeowners is 1.04” etc.

    However, I do agree that the colloquial expression “the average American family has 2 children” does allude to the modal number of children in a family.

    Pardon the pedantry, but the use and abuse of statistics by politicians, lobbyists, advertising agencies, the media and assorted ratbags and drongoes are the bane of people such as I who work in national statistical agencies.

  • Cliff McKeithan

    Max,

    When I hear politicians and their ilk dropping numbers, I recall the saying, “73% of all statistics are made up on the spot.”

  • thebluebird11

    @Cliff,
    Ha! I used that line on someone the other day, someone annoying me with their BS about “half the people here….” or whatever…except I beg to differ, 82% of all statistics are made up on the spot 😉 LOLOL

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