What time means is relative mean 1

Median, mean and frequencies

In this article we clarify the terms median and mean value and focus on the topic of frequencies. You can navigate to your desired topic here:

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Absolute and relative frequency

We look at them Distribution of grades for a class test. We assume that the following notes were written:

\[2;2;2;3;1;5;6;4;5;3\]

This disordered representation is an original list. For a better overview, we will now present this original list in an ordered ranking list:

\[1;2;2;2;3;3;4;5;5;6\]

Next we want to find out the absolute frequency of each note. To do this, we create a table:

We can now see that the bad grade appears twice, for example. So their absolute frequency is two.
\ [\ mathrm {relative \ frequency} = \ frac {\ mathrm {absolute \ frequency}} {\ mathrm {total number}} \]
In the next step we calculate the relative frequency. Basically, the following formula is used to calculate the relative frequency:
A total of 10 students wrote the class work. So our total number is 10.

It is therefore possible to specify the relative frequency as a fraction, as a decimal fraction or as a percentage. The relative frequencies must add up to either 100% or 1.

Daniel explains again the difference between absolute and relative frequency

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Arithmetic mean or mean

When discussing class work, the average grade is often given. We want to calculate the average grade of the present class test. From a mathematical point of view, the average is the arithmetic mean or the mean. When calculating the arithmetic mean, we proceed as follows:
\ [\ overline {x} = \ frac {1 \ times 1 + 2 \ times 3 + 3 \ times 2 + 4 \ times 1 + 5 \ times \ times 2 + 6 \ times 1} {10} = \ frac {33} {10} = 3.3 \]
So the average of the class work is 3.3.

In general, the following formula applies to the arithmetic mean:

\ [\ overline {x} = \ frac {\ mathrm {sum \ of all \ values} \ mathrm {e}} {\ mathrm {number \ of \ values}} \]

Another important value is the so-called mode or mode value. The mode or modal value is the value which has the greatest absolute frequency. In our case, the note good appears the most frequently, namely three times.

So the mode value or mode in our example is: $ x_M = 3 $.

Mean value, arithmetic mean, original list, ranking list, statistics | Math by Daniel Jung

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Median and central value

The median or central value $ \ tilde {x} $ ("x snake") is the value that is in the middle of an ordered ranking list. In order to be able to determine the median, an ordered ranking list must first be created. We now assume that we have the following ordered ranking list:
\[1,\ 2,\ 3,\ 4,\ 5\]
At the beginning we notice that the number of our values ​​is an odd number, namely five. You can see at first glance that the 3 is in the middle or in the center of our ranked list.

The median in this example is: $ \ tilde {x} = 3 $

We also want to look at how to determine the median if the number of our values ​​is an even number. We look at the following ranking list:
\[1,\ 2,\ 3,\ 4,\ 5,\ 6\]
If we look at this ranking, we quickly see that there is no real center or real center from which we can read the median directly. In such cases you look at the two values ​​in the middle and form the arithmetic mean from these two values. In our case, the median would be the mean of the two values ​​3 and 4:
\ [\ tilde {x} = \ overline {x} = \ frac {3 + 4} {2} = \ frac {7} {2} = 3.5 \]

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Stripe, column and pie charts

Data can be presented clearly by using different diagram types. Let’s look at the following example. We assume that a company produces tables in different colors. It produces light gray, medium gray and dark gray tables. During the last production, the following quantities were produced in the different colors:

  • 6 light gray tables
  • 3 medium gray tables
  • 3 dark gray tables

We want to show this division in a strip diagram:
In the left section we see the number of light gray tables, namely six. The middle section shows us the number of medium gray tables, namely three, and the right section shows the number of dark gray tables, also three.
Next we want to illustrate the representation in a column chart (bar chart).
Here the different proportions are shown in separate columns. The $ y $ axis shows the different proportions.
Finally, let's look at the representation in a pie chart:
In a pie chart, the different sectors are divided according to the size of the angle. In order to calculate the individual angle sizes, the respective sectors are seen as parts of a whole circle ($ 360 ^ \ circ $). Our pie chart is based on the following calculations:

  • light gray: $ \ frac {6} {12} \ cdot 360 {} ^ \ circ = 180 {} ^ \ circ $
  • medium gray: $ \ frac {3} {12} \ cdot 360 {} ^ \ circ = 90 {} ^ \ circ $
  • dark gray: $ \ frac {3} {12} \ cdot 360 {} ^ \ circ = 90 {} ^ \ circ $

Sample task

Sample task arithmetic mean

In the 100 m run, the runners achieved the following times:

Find the arithmetic mean (mean or average) and the median.

solution

The arithmetic mean is determined by adding up all occurring values ​​and then dividing by the number of occurring values:

\ begin {align *}
\ overline {x} = \ dfrac {11.21 + 12.54 + 11.76 + 12.32 + 11.91 + 11.99} {6} = 11.955
\ end {align *}

Before the Median (also called central value) can be determined, all values ​​must first be sorted in a ranking list:

\ begin {align *}
11.21 \ quad 11.76 \ quad 11.91 \ quad 11.99 \ quad 12.32 \ quad 12.54
\ end {align *}

The number of dates is even, namely six. In this case, the two values ​​in the middle (or in the center) are added and then divided by two:

\ begin {align *}
\ overline {x} = \ dfrac {11.91 + 11.99} {2} = 11.955
\ end {align *}

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