What's the difference between median and average?

Mean

 The mathematical symbol or notation for mean is &

The mathematical symbol or notation for mean is ‘x-bar’. This symbol appears on scientific calculators and in mathematical and statistical notations.

The ‘mean’ or ‘arithmetic mean’ is the most commonly used form of average. To calculate the mean, you need a set of related numbers (or data set). At least two numbers are needed in order to calculate the mean.

The numbers need to be linked or related to each other in some way to have any meaningful result – for instance, temperature readings, the price of coffee, the number of days in a month, the number of heartbeats per minute, students’ test grades etc.

To find the (mean) average price of a loaf of bread in the supermarket, for example, first record the price of each type of loaf:

  • White: £1
  • Wholemeal: £1.20
  • Baguette: £1.10

Next, add (+) the prices together £1 + £1.20 + £1.10 = £3.30

Then divide (÷) your answer by the number of loaves (3).

£3.30 ÷ 3 = £1.10.

The average price of a loaf of bread in our example is £1.10.

The same method applies with larger sets of data:

To calculate the average number of days in a month we would first establish how many days there are in each month (assuming that it was not a leap year):

Month Days
January 31
February 28
March 31
April 30
May 31
June 30
July 31
August 31
September 30
October 31
November 30
December 31

Next we add all the numbers together: 31 + 28 + 31 + 30 + 31 + 30 + 31 + 31 + 30 + 31 + 30 + 31 = 365

Finally we divide the answer with the number of values in our data set in this case there are 12 (one for each month counted). 

So the mean average is 365 ÷ 12 = 30.42.

The average number of days in a month, therefore, is 30.42.

The same calculation can be used to work out the average of any set of numbers, for example the average salary in an organisation:

Let’s assume the organisation has 100 employees on one of 5 grades:

Grade Annual Salary Number of Employees
1 £20,000 21
2 £25,000 25
3 £30,000 40
4 £50,000 9
5 £80,000 5

In this example we can avoid adding each individual employee’s salary as we know how many are in each category.  So instead of writing out £20,000 twenty-one times we can multiply to get our answers:

Grade Annual Salary Number of Employees Salary x Employees
1 £20,000 21 £420,000
2 £25,000 25 £625,000
3 £30,000 40 £1,200,000
4 £50,000 9 £450,000
5 £80,000 5 £400,000

Next add the values in the Salary x Employees column to find a total: £3,095,000 and finally divide this number by the number of employees (100) to find the average salary:

£3,095,000 ÷ 100 = £30,950.

Quick Tip:

The salaries, in the example above, are all multiples of £1,000 – they all end in ,000

You can ignore the ,000’s when calculating as long as you remember to add them back on at the end.   

In the first row of the table above we know that twenty-one people get paid a salary of £20,000, instead of working with £20,000 work with 20: 

21 x 20 = 420 then replace the ,000 to get 420,000.

Sometimes we may know the total of our numbers but not the individual numbers that make up the total. 

In this example, assume that £122.50 is made by selling lemonade in a week. 

We don’t know how much money was made each day, just the total at the end of the week.  

What we can work out is the daily average: £122.50 ÷ 7 (Total money divided by 7 days). 

122.5 ÷ 7 = 17.50.

So we can say that on average we made £17.50 a day.

We can also use averages to give us a clue of likely future events – if we know that we made £17.50 a day on average selling lemonade in a week then we can assume that in a month we would make:

£17.50 × Number of days in that month

17.50 × 31 = £542.50

We could record average sales figures each month to help us predict sales for future months and years and also to compare our performance.  We could use terms like ‘above average’ – to refer to a time period when sales were more than the average amount and likewise ‘below average’ when sales were less than the average amount.

Using speed and time as data to find the mean:

Using speed and time as data to find the mean:

If you travel 85 miles in 1 hour and 20 minutes, what was your average speed?

The first thing to do with this problem is to convert the time into minutes – time does not work on the decimal system as there are 60 minutes in an hour and not 100.  Therefore we need to standardise our units before we can start:

1 hour 20 minutes = 60 minutes + 20 minutes = 80 minutes.

Next divide the distance travelled by the time taken: 85 miles ÷ 80 minutes

85 ÷ 80 = 1.0625.

Our average speed therefore was 1.0625 miles per minute.

Convert this figure back to hours by multiplying by 60 (the number of minutes in an hour). 

1.0625 × 60 = 63.75mph (miles per hour).

For Spreadsheet users:

Use the <average> function to calculate the mean average in a spreadsheet. The following example formula, assumes your data is in cells A1 to A10:

=average(A1:A10)

Heronian Mean

Used in geometry to find the volume of a pyramidal frustum. A pyramidal frustum is basically a pyramid with the tip sliced off.

Video

The median

Let’s move now to the median, probably less known

Definition

A value in an ordered set of values below and above which there is an equal number of values or which is the arithmetic mean of the two middle values if there is no one middle number

The median is more a statistic term than mathematic

The goal of a median is to find the central point in a set of valuesWhen found, you must have as many upper than lower values

Usage

We generally use the median when the values aren’t evenly distributed

For example, it’s the case for salariesAn average salary doesn’t make sense, the median salary is a better informationFor example, if you include unemployed and billionaires, what the real sense of the average?

That’s why the median is better. The median excludes the extremes, to find the central point. That is more relevant, mainly for big sets of valuesI will show you that in the next example

Example

Here is an example with different values. I will show you the result for average and median on this setLet’s say we have 5 peoples, here are their salaries:

  • Anitha: $1,500
  • Robert: $1,800
  • Raymond: $1,000
  • Jack: $25,000
  • Bill: $350,000

I’ll not do the math, as tools like Microsoft Excel allow you to make it directly. You can also use a scientific calculator(In this case, there is no real calculation to do, but hey I don’t spoil you ^^)I’ll go directly to the result, and explain it to you just after:

So, I just type the same values in Excel
With the

So, I just type the same values in ExcelWith the results above, you can see that the average salary is $75,860But there is no sense, as nobody except Bill get this salary levelSo it’s not a good idea of the reality

But the median salary tells us the central point, at $1,800As the set of values is small, it’s not really a better idea of the situation, but this result is still much closer to realityWe have 2 people over 1800, and 2 under 1800, so it’s exactly what we wanted

Examples

A few practical examples:

  • In order to report on full resolution time, use the Explore Full Resolution Time (hrs) [MED] metric. Choose the median operator because a number of tickets have been under investigation for a while and these tickets might skew your report.
  • To check the average amount of replies posted by the agents use the # Replies [MED] metric because the number of replies is more or less constant.
  • To figure out how fast the support team replies to new requests, use the Explore First Reply Time (hrs) [MED] metric. Since first reply time is normally constant, create a metric that will count the average first reply time. Additionally, you can filter out proactive tickets, created by agents from the report because they might have a high first reply time.

Note:  For changing the metric aggregator in Explore, see Choosing metric aggregators. 

 

How to find the mean median mode by hand: Steps

How to find the mean median mode: MODE

  • Step 1: Put the numbers in order so that you can clearly see patterns. For example, lets say we have 2, 19, 44, 44, 44, 51, 56, 78, 86, 99, 99. The mode is the number that appears the most often. In this case: 44, which appears three times.

How to find the mean median mode: MEAN

  • Step 2: Add the numbers up to get a total. Example: 2 +19 + 44 + 44 +44 + 51 + 56 + 78 + 86 + 99 + 99 = 622.  Set this number aside for a moment.
  • Step 3: Count the amount of numbers in the series. In our example (2, 19, 44, 44, 44, 51, 56, 78, 86, 99, 99), we have 11 numbers.
  • Step 4: Divide the number you found in step 2 by the number you found in step 3. In our example: 622 / 11 = 56.5454545. This is the mean, sometimes called the average.

Dividing the sum by the number of items to find the mean.

How to find the mean median mode: MEDIAN

If you had an odd number in step 3, go to step 5. If you had an even number, go to step 6.

  • Step 5: Find the number in the middle of the list. This is the median. 2, 19, 44, 44, 44, 51,56, 78, 86, 99, 99.
  • Step 6: Find the middle two numbers. For example, 1, 2, 5, 6, 7, 8, 12, 15, 16, 17. The median is the number that comes in the middle of those middle two numbers (7 and 8), so that number would be 7.5 in this case. In order to do this mathematically, add the two numbers together and divide by 2.

Tip:  You can have more than one mode. For example, the three modes of 1, 1, 5, 5, 6, 6 are 1, 5, and 6.

Like the explanation? Check out the Practically Cheating Statistics Handbook, which has hundreds more step-by-step solutions, just like this one!

What does the median show vs. the mean?

"Mean" and "average" are synonyms in this context. The difference between what the mean shows and what the median shows is the same as the difference between median and average. The median is the figure at which half of the data points fall above and half fall below. The mean (or "average") is the sum of all data divided by the number of data points.

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