Does the order I type the numbers change the median?

The typed order does not matter, because the median is defined on the ordered list. Sorting is part of the procedure, not an optional tidy-up. What does matter is leaving a number out or typing the same invoice twice. Either mistake shifts the middle position. Keep a count of how many values you intended, and match it to the count after you sort, before you trust the center.

Why is the median sometimes not one of my numbers?

With an even count, the two middle values are averaged, and that average may sit between them. A list whose two centers are 15 and 16 has a median of 15.5, even though 15.5 never appears. That is correct. Do not round it back to a typed value unless your report's rounding rule says so. Tell the reader it is the midpoint of the two central observations.

Should groups of different sizes share one median?

A median of group medians treats a group of 8 like a group of 800. That is rarely what you want. Pool the raw observations and find the median of the pooled list if every observation should count once. If you cannot pool the rows, say you are summarizing group medians, and give each group's size so the reader sees the imbalance you could not fix.

Is the median always a better summary than the mean?

No. The median describes the middle position, and it can ignore a total you are required to fund. Bills, minutes, and points that must add back to a known sum often need the mean or the sum itself. Choose the median when a long tail would distort a typical case. Choose the mean when every unit of size has to count in the result. Keep the unit, the count, and the name of the formula in the same sentence as any figure you copy into a report.

What if several values tie at the middle?

Ties are allowed and simply sit next to each other after the sort. If the middle position lands on a tied value, that value is the median. You do not break the tie by averaging in a neighbor from outside the center. Report the tie as well, for example that 6 of 11 scores equal the median, so the reader sees a stack rather than a single typical paper.

When the Median Tells the Truth the Mean Will Not

Five wages and one large rate

Sort hourly rates of 62, 16, 19, 17, and 18 into 16, 17, 18, 19, and 62. The middle position is the third rate, 18 dollars. The mean of the same five numbers is 26.40, because the sum is 132. A job advert that quotes 26.40 as typical pay is describing the pull of the 62, not the center of the list. Quote 18 when the question is what a worker in the middle of this group earns. The range still belongs in the sentence, because 18 does not warn anyone that one rate is 62.

Move the 62 up to 90 and the ordered list becomes 16, 17, 18, 19, 90. The median remains 18. The sum is now 160 and the mean rises to 32. The middle paper did not notice the change at all. That silence is the point of using a median on a skewed pay list. It is also the reason a median can hide a change you care about, such as a new top rate. If the top of the scale is the news, report the maximum beside the median, not the median alone.

Even counts use two middle values

Six response times in minutes, already ordered, are 4, 8, 15, 16, 23, and 42. There is no single middle seat. The two central positions are 15 and 16, and their average is 15.5 minutes. That 15.5 is the median, not a mean of all six. The mean of the whole list is 18, because the sum is 108. People often grab one of the two central values and stop. Take both, average only those two, and label the result as the median of an even count so the half-minute is explained.

The Median Calculator sorts a copy of the list and, on an even count, averages only the two central values. On this list it returns 15.5, and the 42 never enters that average. If you need the equal share of the 108 minutes, you want the mean of 18 instead, and you should say so. A staffing plan that uses 15.5 as if it were the total divided by six will understate the time the whole set actually consumed. Match the summary to the question: a typical stop, or a share of the total.

Sale prices and the middle house

Five sale prices in thousands are 220, 240, 260, 280, and 900. Ordered, the middle price is 260. The sum is 1,900, so the mean is 380. A headline average of 380 sits above four of the five sales. A buyer using 380 as the typical house in this handful will aim far above the center of the market these five sales describe. Report 260 as the median price and mention that the top sale was 900. The mean is still useful if you are totaling revenue, because the 900 is real money.

Add a sixth sale at 250. The ordered list is 220, 240, 250, 260, 280, and 900. The two middle prices are 250 and 260, so the median is 255. The mean uses the sum 2,150 divided by 6, which is about 358. One extra ordinary sale barely moved the median and pulled the mean down from 380 toward 358, and the 900 is still doing most of the pulling. When a new sale comes in, update both numbers. If you update only the mean, a reader cannot tell a cooler typical sale from a smaller mansion.

A low outlier drags the mean down

Ordered scores of 4, 18, 19, 21, and 22 have a median of 19. The sum is 84 and the mean is 16.8. The low score of 4 sits 15 points under the median and pulls the equal share below every score except itself. A report that says the group averaged 16.8 makes the room sound weaker than four of the five papers. If 4 is a real score, keep it in the mean and show 19 as the median. If 4 was a blank coded as zero by mistake, fix the code before either summary is discussed.

Compare the damage of a low spike with a high one on a list this short. Replacing 4 with 40, and keeping 18, 19, 21, and 22, gives the ordered list 18, 19, 21, 22, 40. The median is 21. The sum is 120 and the mean is 24. In both versions one odd score moves the mean by several points and moves the median by only the shift from 19 to 21, which happened because the odd score changed which value sat in the middle, not because 40 was added into the median. Say which end of the list is strange.

When the total still needs a mean

Medians resist tails, and that resistance is a drawback when the total is the decision. Five invoices of 40, 42, 45, 48, and 200 dollars have a median of 45 and a mean of 75, since the sum is 375. If you budget 45 dollars times 5, you set aside 225 and you are 150 dollars short of the real total. The 200 is not a nuisance to be medianed away. It is part of the cash. Use the mean, or better, use the sum, when you are covering every bill. Use the median when you are describing a typical bill.

A practical rule is to compute both and write the one your verb requires. Typical, middle, and usual point toward the median on a skewed list. Total, share, per person from a fixed pot, and rate that must add back to a sum point toward the mean. On the invoice list, say the typical bill is 45 dollars and the average bill, as a share of 375, is 75. Readers who hear only one of those sentences will make the wrong plan. Two numbers take one extra line and remove a common budget miss.

A sentence that states the middle clearly

Name the count, the median, and whether the count was odd or even. For the six response times, write that the median of six times is 15.5 minutes, the average of the two middle values 15 and 16. For the five wages, write that the median of five rates is 18 dollars, the third when ordered from 16 to 62. That wording stops a colleague from re-averaging the whole list and calling the result a median. It also shows that 15.5 was not a typed observation. It was the midpoint of two typed observations.

Keep the original order somewhere else, because the median's sort is a copy. A timeline of response times may need the sequence 42 first and 4 last, and sorting for the median throws that sequence away. Store the raw list, then store the median beside it. If two people compute different medians, the usual causes are an unsorted middle, an even count that used only one of the two centers, or a blank turned into zero. Those three checks solve most disagreements before anyone argues about the data. Keep the unit, the count, and the name of the formula in the same sentence as any figure you copy into a report.

Open the worksheet

The matching page is the Median Calculator. It opens with a worked example, names the formula, and keeps the steps next to the result. Use it when you want the arithmetic on your own numbers rather than the argument on this page.