Mean Median Mode Calculator

Mean = Σ xᵢ ÷ n Median = the middle value once the data is sorted Mode = the value with the highest frequency

xᵢ

One value in the data set. Every value counts once, whether it is near the middle or out at an extreme

n

How many values there are, up to 200 here. All three of these centers divide by the count rather than by n − 1, so there is no sample-and-population choice to make on this page

x̄

The mean. It is the only one of the three that every single value can move, which is why one extreme value drags it and leaves the other two alone

M

The median: sort the values and take the middle one, or the average of the two middle ones when the count is even. Half the data lies below it and half above

f

The frequency of the mode — how many times the most common value appears. A frequency of 1 means no value repeats, and then every value is tied for the title

Quote the median when the list has outliers or a long tail — house prices, incomes, waiting times, anything where one value can be enormous. The median is unmoved by how far out an extreme value sits, only by how many values are on each side. Quote the mean when every value should carry the same weight and the total matters: an average score, an average spend per customer, a rate. Quote the mode when the question is what is typical in the sense of most common — the most frequent shoe size to stock, the most common number of passengers. If the three agree, the data is roughly symmetric and the choice does not matter. If they disagree, report the median and say why: a right-skewed set has the mode below the median and the mean above it, and the mean is the one being pulled.

1. A skewed week: the three centers pull apart

   1. Sorted: 1, 2, 2, 3, 4, 7, 9 — seven values, so the middle one is the fourth
   2. Median: 3
   3. Sum: 1 + 2 + 2 + 3 + 4 + 7 + 9 = 28, so the mean is 28 ÷ 7 = 4
   4. Mode: 2 appears twice and nothing appears three times, so the mode is 2 with a frequency of 2

   Read the three answers as a shape rather than as three estimates of one thing: mode 2, median 3, mean 4, in that order, is the signature of a right-skewed set — the 7 and the 9 have pulled the mean up past the median, while the mode sat still. If the 9 were 90 the mean would jump to 15.6 and the median would not move at all.

2. Two modes, and no obvious middle

   1. Sorted: 2, 2, 5, 5, 9 — the middle value is 5
   2. Sum: 23, so the mean is 23 ÷ 5 = 4.6
   3. Two values appear twice each: 2 and 5, so there are two modes and the panel reports the smaller one

   This data set is bimodal — two values tie for most frequent — and a data set like this is often two groups accidentally pooled together, in this case a cluster of 2s and a cluster of 5s. The page reports the smaller of the tied modes and the count of modes beside it, which is the honest reading: the single number 2 is not more of a mode than 5, it is just the first one in order.

3. No value repeats at all

   1. Every value appears exactly once, so nothing is more frequent than anything else
   2. Frequency of the mode: 1; number of modes: 3, which is every value in the set
   3. Mean and median: both 2

   A frequency of 1 alongside a mode count equal to the number of values is how this page says "there is no mode" — and that is a more useful statement than an empty answer, because three separate numbers together tell you every value tied. The mode column still shows a value, the smallest, purely so the row is not blank.

All three of these describe the middle, and the range printed beside them says only how far apart the two outermost values are — which is not the same as how the data is spread between them: two data sets can share a mean, a median and a range and be nothing alike, one tightly clustered and one spread from end to end. A skewed set is better described by the median and the quartiles than by the mean, because the mean is dragged toward the tail while the median is not. The mode is the least stable of the three — it can jump to a different value when a single entry changes, and with continuous measurements it often has no useful mode at all, so what looks like a mode may just be rounding. This page reads plain numbers, so a frequency table has to be expanded into its values before it can be used here, and the list is capped at 200 values.

How do I find the mean, median and mode?

Add the values and divide by how many there are for the mean. Sort them and take the middle one for the median — with an even count, average the two middle values. Count how often each value appears for the mode. For 1, 2, 2, 3, 4, 7, 9 the sum is 28 over 7 values, so the mean is 4; the fourth of seven sorted values is 3, so the median is 3; and 2 appears twice, so the mode is 2.

Which one should I use?

Use the median when there are outliers or a long tail, because it does not care how far out an extreme value sits — only how many values are on each side of it. Use the mean when every value should count equally and the total is what matters, such as an average score or a rate. Use the mode when you want the most common value rather than the middle one, as when deciding which size to stock. The three agreeing is a useful signal in itself: it means the data is close to symmetric and the choice does not matter much.

What does it mean when there is no mode?

It means every value appears the same number of times, so none of them is more frequent than the rest. This page says so rather than leaving the row blank: the frequency of the mode reads 1 and the number of modes equals the count of values, so 1, 2, 3 reports three modes. That is not an error — with a small list and no repeats it is the expected answer, and with continuous measurements it is the usual one.

Can a data set have more than one mode?

Yes, and it is common. If two values tie for the highest frequency, both are modes and the data set is called bimodal; the panel reports the smaller of them and shows the number of modes next to it. Two modes often mean two groups have been pooled by accident — a cluster of one size and a cluster of another — which is worth noticing before quoting a single center for the whole list.

Is there a sample and population version of these?

No, and this page deliberately has no drop-down for it. All three of these centers divide by the count of values; the n − 1 correction belongs to the spread, not to the middle. A mean is a mean whether the list is a sample or the whole group. If you need the sample or population difference, that is the standard deviation and variance pages, which do have the choice.

• Measures of the Center of the Data — Introductory Statistics 2e, section 2.5 — OpenStax, Rice University
• Skewness and the Mean, Median, and Mode — Introductory Statistics 2e, section 2.6 — OpenStax, Rice University