# Some Presentations of Data

The collection, organisation and analysis of numerical information are all subject called **statistics**. Pieces of numerical and other information are called **data**. **Data** is ‘a series of facts from which conclusions may be drawn’.

In order to collect data you need to observe or to measure some property. This property is called variable.

## Tables

Let’s look at this data set:

13, 11, 10, 14, 14, 11, 9, 7, 22, 20, 24

What conslusion can we draw? It doesn’t mean anything to us.

But if we see table below we can get some clue that this is data about soccer games of German clubs in 2017 and data is in column AS. We still don’t know what data means because we don’t know what column AS means.

Div | Date | HomeTeam | AwayTeam | FTHG | FTAG | FTR | HTHG | HTR | HS | AS | B365H |
---|---|---|---|---|---|---|---|---|---|---|---|

D1 | 18/08/17 | Bayern Munich | Leverkusen | 3 | 1 | H | 2 | 0 | H | 13 | 19 |

D1 | 19/08/17 | Hamburg | Augsburg | 1 | 0 | H | 1 | 0 | H | 11 | 2.1 |

D1 | 19/08/17 | Hertha | Stuttgart | 2 | 0 | H | 0 | 0 | D | 10 | 2 |

D1 | 19/08/17 | Hoffenheim | Werder Bremen | 1 | 0 | H | 0 | 0 | D | 14 | 1.75 |

D1 | 19/08/17 | Mainz | Hannover | 0 | 1 | A | 0 | 0 | D | 14 | 2 |

D1 | 19/08/17 | Schalke 04 | RB Leipzig | 2 | 0 | H | 1 | 0 | H | 11 | 2.8 |

D1 | 19/08/17 | Wolfsburg | Dortmund | 0 | 3 | A | 0 | 2 | A | 9 | 3.8 |

D1 | 20/08/17 | Freiburg | Ein Frankfurt | 0 | 0 | D | 0 | 0 | D | 7 | 2.45 |

D1 | 20/08/17 | M’gladbach | FC Koln | 1 | 0 | H | 0 | 0 | D | 22 | 1.91 |

D1 | 25/08/17 | FC Koln | Hamburg | 1 | 3 | A | 0 | 2 | A | 20 | 1.95 |

D1 | 26/08/17 | Augsburg | M’gladbach | 2 | 2 | D | 1 | 2 | A | 24 | 3.25 |

But I also have this information:

- Div = League Division
- Date = Match Date (dd/mm/yy)
- HomeTeam = Home Tea
- AwayTeam = Away Team
- FTHG and HG = Full Time Home Team Goals
- FTAG and AG = Full Time Away Team Goals
- FTR and Res = Full Time Result (H=Home Win, D=Draw, A=Away Win)
- HTHG = Half Time Home Team Goals
- HTR = Half Time Result (H=Home Win, D=Draw, A=Away Win)
- HS = Home Team Shots
**AS = Away Team Shots**- B365H = Bet365 home win odds

So now we know that 13, 11, 10, 14, 14, 11, 9, 7, 22, 20, 24 are Shots that Away team made in game. So 13 is shots (to goal) of Leverkusen in game against Bayern Munich on 18/08/17, 11 is number of shots that Ausburg make in game against HAmburg on 19/08/17.

# Types of data - variables

A variable is **qualitative** if it not possible to take a numerical value(Leauge Division, Date,Team name, Full Time Result)
A variable is **quantitative** if it can take numerical value (Full Time Home Team Goals,Away Team Shots). Quantitative variable can be **continuous** or **discrete**. Continuous variable can take any value in given range (Bet365 home win odds). Discrete variable has clear steps between possible values Full Time Away Team Goals can’t take value 1.25.

As You can see presenting data in table provides a lot of information. Data in table tells a lot of, but this data is fine-grained. What if we want to know total or average team shots in season this is not right type of presentation.

# Mean Median Mode and Range

Let us make some digression and explain what is Mean, Median and Mode.

Data set: 4, 5, 3, 8, 2, 2, 10, 9

**Mean(or average)** is number that you get when add all data points and divide by the number of data points. (4 + 5 + 3 + 8 + 2 + 2 + 10 +9)/8 = ^{43}⁄_{8} = 5.375

mean=sum of data# of data points

**Median** is the middle number, found by ordering all data points and picking out the one in the middle (or if there are two middle numbers, taking the mean of those two numbers).
4 and 5 (two middle numbers)
^{9}⁄_{2} = 4.5.
For example if number of variables in data set is even 4, 5, 7, 12, 22 then mean is number in middle of ordered data set 7.

**Mode** is the most frequent number, the number that occurs the highest number of times, in case of our example data set 2. There can be no mode (1,2,3,4) Mode = {}, one mode(1,1,2,3,4) Mode = {1}, or multiple modes(1,1,2,2,3,4,5) Mode = {1, 2} in a dataset.

**Range** The largest value in the list is 10, and the smallest is 2, so the range is 10 - 2 = 8 (max - min).

## Steam-and-leaf

One interesting presentation is steam-and-leaf

Suppose that a height of children in one class of elementary school is measured and data set is:

65, 54, 76, 72, 71, 83, 72, 92, 60, 42, 73, 84, 72, 97, 72, 91, 53, 63, 84, 85

Let us sort data and take the leading digit (10’s digit) as stem, and trailing digits are leafs.

42, 53, 54, 60, 63, 65, 71, 72, 72, 72, 72, 73, 76, 83, 84, 84, 85, 91, 92, 97

Steam | Leaf |
---|---|

4 | 2 |

5 | 34 |

6 | 035 |

7 | 1222236 |

8 | 3445 |

9 | 127 |

4|2 means

5|34 means 53 and 54

If You turn this on side you can see the “shape” of the height distribution and all data is visible (maintained).

This is called ordered steam-and-leaf because leaf are ordered |7|**1222236**.

Depending of data set stem can be taken as two digits. 164 184 186 175 187

16|4

17|5

18|467

If data set is large stem-and-leaf diagram is not best method of displaying data and you need to use other methods.

## Tally

For large data sets of data you may wish to divide the data into groups, called **classes**.

We can use tally to present data. For example favorite drinks of 62 students can be presented like this.

## Histogram

Data is collected from big food shoop from all types of cerelars and amout of sodium is in cerelars is variable. Data is presented in next table as grouped frequency distribution. [0-49], [50-99] … are classes

Amount of sodium (mg) | Tally | Frequency |
---|---|---|

0-49 | ***** ***** ** | 12 |

50-99 | ***** | 5 |

100-149 | ***** ***** ** | 12 |

150-199 | ***** ***** ***** ** | 17 |

200-249 | ***** ***** ***** ***** * | 21 |

250-299 | ***** **** | 9 |

300-349 | * | 1 |

There is no clear rule about how many classes should be chosen or what size they should be, but it is usual to have 5-10 classes.

Grouping data into classes means losing some information. We can see in table that there are 5 types of cereals with amount of sodium between 50 and 99 but we don’t know exact values of observations.

Amount of sodium is **continuous variable** and amount is rounded to the nearest mg. Class labeled 50-99 would contain values from 49.5 up to (but not including) 99.5. These real endpoints 49.5 and 99.5 are referred as **class boundaries**. 49.5 <= Class 50-99 < 99.5 or [49.5-99.5)

When a grouped frequency distribution contains continuous data, one form of display is the **histogram**.

Histogram looks similar to a bar chart.

A chart which represents continuous data is a **histogram** if
- the bars have no space between them (they mey be bars of height zero)
- the area of each bar is proportional to the frequency

widht-of-class * height = frequency

height = frequency / widht-of-class

Height is known as **frequency density**

Time is up, for this post :)