# What is 9 2 7 21

### Who has more likes?

Tony and Carla are new to Instagram. You write down the number of likes every day.

Here is the frequency table:

Days | Mon | Tuesday | Wed | do | Fr. | Sat | So |
---|---|---|---|---|---|---|---|

Tony | |||||||

Carla |

So many numbers ... is there any way to sum that up?

You have probably already dated **average** belongs. So that you can compare the data better, you could ask: Who gets more likes per day on average?

You calculate the average by adding up all the likes and dividing the result by the number of days.

Result for Tony:

$$ frac {2 + 4 + 3 + 5 + 1 + 4 + 2} {7} = frac {21} {7} = 3 $$

Result for Carla:

$$ frac {1 + 2 + 4 + 3 + 2 + 2 + 7} {7} = frac {21} {7} = 3 $$

So Carla got as many likes on average as Tony during the week.

### The arithmetic mean

The word "average" is more colloquial. Mathematicians call this the average **arithmetic mean** or short **Average**.

The symbol is often $$ bar x $$. Say: "x across".

You use the arithmetic mean to mark confusing data with only one value. This is a good way of comparing the data.

How to calculate the arithmetic mean:

**Step 1:** Add up all the dates.**2nd step:** Divide the result by the number of dates.

You calculate the arithmetic mean $$ bar x $$ with:

$$ bar x = frac {sum of \ all \ data} {number \ of \ data} $$

### Body sizes

Three girls and three boys from 6c indicated their height:

girl | 1.23 m | 1.45 m | 1.25 m |

Boys | 1.05 m | 1.34 m | 1.35 m |

Give the arithmetic mean of the girls and boys.

Option 1: do the math in 2 steps.**Girl:****Step 1:** $$ 1.23 \ m + 1.45 \ m + 1.25 \ m = 3.93 \ m $$ **2nd step:** $$ frac {3.93 \ m} {3} = 1.31 \ m $$

Result: The average height of the 3 girls is $$ 1.31 \ m $$.

Option 2: summarize both steps. **Boys:**

$$ frac {1.05 \ m + 1.34 \ m + 1.35 \ m} {3} $$

$$ = frac {3.74 \ m} {3} $$

$$ = 1.2466 ... \ m $$

$$ approx1.25 \ m $$

Result: The average height of the 3 boys is about $$ 1.25 \ m $$.

In the case of division, you may have to round. Round off the digits 0, 1, 2, 3 and 4. With the digits 5, 6, 7, 8, 9 you round up.

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### Mean and absolute frequencies

There are some values in the following frequency list **multiple** in front:

value | 1,22 € | 1,46 € | 2,10 € | 4,82 € |

absolute frequency |

To calculate the mean for this case, do the following:

**Step 1:** Multiply each value by its absolute frequency.**2nd step:** Add up the products.**3rd step:** Divide the result by the sum of the absolute frequencies.

$$ frac {2 * 1.22 \ € + 3 * 1.46 \ € + 1 * 2.10 \ € + 4 * 4.82 \ €} {2 + 3 + 1 + 4} = frac {28, 20 \ €} {10} = 2.82 \ € $$

The mean value is $$ 2.82 \ € $$.

### The arithmetic mean in the diagram

These bar charts show average temperatures in Germany for the years 2013 and 2015 over the seasons winter, spring, summer and autumn.

Calculate the mean of these averages. Round to one decimal place.

**Year 2013:**

$$ barx = frac {0.3 + 6.7 + 17.7 + 9.5} {4} = frac {34.2} {4} approx8.6 \ ^ ° C $$

The mean value of the average temperatures in 2013 is about $ 8.6 \ ^ ° C $.

**Year 2015:**

$$ barx = frac {1.9 + 8.6 + 18.4 + 9.6} {4} = frac {38.5} {4} approx 9.6 \ ^ ° C $$

The mean value of the average temperatures in 2015 is about 9.6 \ ^ ° C $$.

The 2 mean values are shown here. This gives you an even quicker overview of the data for the seasons.

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