2.2.4. Histogram
Histograms are frequency diagrams to a data set, where the absciss axis or horizontal axis or 'X' axis represent the range of response variable. Into ordinates axis or vertical axis or 'Y' axis we have the value frequency of this variable.
The most popular histograms are the representative gaussian or normal curves and have a characteristic bell-shaped. Histograms usually use data grouping to compose its classes or ranges.
The histograms are graphical displays of a variable behavioring, also can help on this behavior explanation. Through this graphics comes immediatelly the identification of mode's value or the modal class from the sample or population.
Simmetry measure how skewness and measure of the "peakedness" kurtosis also are visible in a histogram.
Other convenient information in histograms is in the cumulative frequency, which give the probability density functions.
2.2.4.1. Histogram Command
Access:
- Menu - Metrixus | Histogram
 -Metrixus Toolbar
Description:
Create a histogram like graphic to the cells interval selected. Allow sample (n-1) or populational (n) calculus and the inclusion of means and standard deviation for each histogram range. Through normalization option, its possible to create histograms to probability density function modeling. This command creates a new file which contain the results.
The data region or interval must be a contiguous interval whith at least two numeric fields. Fields with text format or empty will be disconsidered. The data interval must be selected before using this command.
The max number of ranges to the histogram depends of data amount, not exceeding 50 ranges (or classes). The default are 15 classes or half of data amount, or which is the lesser.
Histograms are used to represent data graphically, but when this data are prices or quotation of a financial asset, its interesting plot the histogram of this assets returns. Hypothesis about the type of distribuction of an asset return are based in this histograms, so its utility. From the option Histogram of returns from asset prices, all analised data are considered as assets quotations and transformed in returns through the time, musting data, so, sorted by time.
Important:
In the histogram of returns, to the determination of statistics parameters of data - like media and standard deviation - is not aplied in any logari operator logarithmic to the quotation return. By this, the mean presented must be understood as ahope for returns and not the mean return, as well the standard deviation must be understood as volatileness of the returns hope!
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Important:
Also to the returns histogram, the data must be ordered. If exist more than one collumn in the cells interval, the data must be in order inside the lines and collumns. Any data from collumn A comes before any from collumn B! Data in line 1 from collumn A comes before from data in line 2 and collumn A!
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Generating graphics or worksheets whithout colors allow an easy data printing hereafter represent better running performance.
The result if generating a hsitogram its a new worksheet with the statistics data, this is, without any connection with the 'database' which generate the result. This worksheet will have the following informations, where n is the valid data amount (or amount -1 for the return histogram)
- Mean: mean of data.
- Median: center value of data. If it have two central values, will show the arithmetic mean of these values.
- Mode: mean value of the range or class whith bigger frequency or occurrence. If exist more than one mode, show empty.
- Maximum: maximum data value.
- Minimum: data minimun value. Text and empty cells are disconsidered.
- St. Dev.: standard deviation. If exist only one data in range, shows empty.
- Population:
- Sample:
- Skewness: data simmetry measure. So, a negative measure indicates data moved to the right and a positive measure indicates data moved to left. If n is less than 3, shows empty.
- Population:
- Sample:
- Exc. Kurt.: measure of the "peakedness" of data in relatinh with normal curve (kurtosis=0). Also knows as excess kurtosis. So, a negative measure indicates "peakedness" relating the normal curve and a positive indicates "peaks" relating to the normal. If n less tha 4, shows empty.
- Population:
- Sample:
The option of histogram normalization result in a second histogram where the frequency of each range or class its divided by the data amount and ranges width. This way, a histogram with total area (integral) equals 1 is obtained. A cumulative frequency line also its showed and, in function of normalization, corresponds to the function of cumulative probability density. Although be less intuitive represent the histogram normalization using this way on contrary percentuals or proportions in each range, this normalization its indicated to draw functions of probability density.
The option of indicators for ranges adds to the histogram a line which represent the standard deviation of each range, allowing to analyse the data dispersion inside the classes.
Usage sample with data:
Data Histogram – parameters:
- Data interval D7:M16 (100 data)
- Ranges (or classes) 15
- Sample
- Include statistics indicators
- Add normalized histogram
- Color graphics
- Data
| 7.92713 | 6.71029 | 10.25132 | 12.07287 | 9.22936 | 9.74868 | 8.65102 | 10.77064 | 11.34898 | 13.28971 |
| 8.01109 | 6.89046 | 10.30194 | 12.16064 | 9.28308 | 9.79913 | 8.71331 | 10.82493 | 11.41260 | 13.50137 |
| 8.09167 | 7.04842 | 10.35275 | 12.25278 | 9.33629 | 9.84946 | 8.77437 | 10.87983 | 11.47769 | 13.76158 |
| 7.83936 | 6.49863 | 10.20087 | 11.98891 | 9.17507 | 9.69806 | 8.58740 | 10.71692 | 11.28669 | 13.10954 |
| 8.24421 | 7.31849 | 10.45509 | 12.45306 | 9.44136 | 9.94986 | 8.89323 | 10.99170 | 11.61284 | 14.65268 |
| 7.74722 | 6.23842 | 10.15054 | 11.90833 | 9.12017 | 9.64725 | 8.52231 | 10.66371 | 11.22563 | 12.95158 |
| 8.16927 | 7.18985 | 10.40379 | 12.34998 | 9.38904 | 9.89969 | 8.83432 | 10.93540 | 11.54439 | 14.10750 |
| 7.54694 | 5.34732 | 10.05014 | 11.75579 | 9.00830 | 9.54491 | 8.38716 | 10.55864 | 11.10677 | 12.68151 |
| 8.31676 | 7.43690 | 10.50669 | 12.56310 | 9.49331 | 10.00000 | 8.95120 | 11.04880 | 11.68324 | 6.71029 |
| 7.65002 | 5.89250 | 10.10031 | 11.83073 | 9.06460 | 9.59621 | 8.45561 | 10.61096 | 11.16568 | 12.81015 |
Results:
| Sample | 100 |
| Mean | 9.967 |
| Median | 9.975 |
| Mode | 10.000 |
| Maximum | 14.653 |
| Minimum | 5.347 |
| St. Dev. | 1.949 |
| Skewness | 0.002 |
| Exc. Kurt. | -0.363 |
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| Ranges | <= Inf. | < Sup.* | Freq. | Range Mean | Range St. Dev. | | Norm. | Freq. N. | Accum. |
1 | 5.35 | 5.97 | 2 | 5.62 | 0.39 | | 1 | 0.032 | 0.032 |
2 | 5.97 | 6.59 | 2 | 6.37 | 0.18 | | 2 | 0.032 | 0.064 |
3 | 6.59 | 7.21 | 5 | 6.91 | 0.21 | | 3 | 0.081 | 0.145 |
4 | 7.21 | 7.83 | 5 | 7.54 | 0.17 | | 4 | 0.081 | 0.226 |
5 | 7.83 | 8.45 | 8 | 8.12 | 0.19 | | 5 | 0.129 | 0.355 |
6 | 8.45 | 9.07 | 11 | 8.77 | 0.20 | | 6 | 0.177 | 0.532 |
7 | 9.07 | 9.69 | 11 | 9.39 | 0.17 | | 7 | 0.177 | 0.709 |
8 | 9.69 | 10.31 | 13 | 10.00 | 0.20 | | 8 | 0.210 | 0.919 |
9 | 10.31 | 10.93 | 11 | 10.61 | 0.17 | | 9 | 0.177 | 1.096 |
10 | 10.93 | 11.55 | 11 | 11.23 | 0.20 | | 10 | 0.177 | 1.273 |
11 | 11.55 | 12.17 | 8 | 11.88 | 0.19 | | 11 | 0.129 | 1.402 |
12 | 12.17 | 12.79 | 5 | 12.46 | 0.17 | | 12 | 0.081 | 1.483 |
13 | 12.79 | 13.41 | 4 | 13.04 | 0.21 | | 13 | 0.064 | 1.547 |
14 | 13.41 | 14.03 | 2 | 13.63 | 0.18 | | 14 | 0.032 | 1.580 |
15 | 14.03 | 14.65 | 2 | 14.38 | 0.39 | | 15 | 0.032 | 1.612 |
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*To the last range <= Sup.
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Histogram containing range indications and the standard deviation of each range.
Histograma= Histogram ; Ocorrências= Occurrences ; Faixas= Ranges
DP/Faixas= St. Dev./Ranges ; Freq.= Frequency ; DP Fx= St. Dev. Fx
Normalized Histogram.
The cumulative frequency line (left axis) represent the cumulative probability function and the area above the line is 1, considering the range width equal to the maximum range value minus the minimum value divided by the range amount.
Histograma Normalizado= Normalized histogram ; Freq. Normalizadas= Normalized Frequencies
Faixas= Ranges ; Freq. N.= N. Freq. ; Acum=Accumulated
Usage sample with quotations:
Histogram of quotations returns - parameters:
- Data interval C18:M37 (220 quotations)
- Ranges (or classes) 15
- Sample
- Include statistics indicators
- Add normalized histogram
- Graphics in black and white
- Generate histogram of returns
- Data – quotations of an asset in time ordered by lines and after by columns
| 58.20 | 58.00 | 47.00 | 46.50 | 44.00 | 46.00 | 57.50 | 57.00 | 35.50 | 34.01 | 41.30 |
| 59.90 | 58.00 | 48.50 | 45.50 | 41.50 | 46.00 | 57.50 | 53.80 | 35.50 | 36.31 | 41.30 |
| 61.20 | 52.00 | 48.00 | 43.00 | 44.50 | 46.00 | 56.50 | 52.70 | 32.50 | 37.00 | 40.50 |
| 61.00 | 47.00 | 48.55 | 45.50 | 42.50 | 47.00 | 55.00 | 53.80 | 34.60 | 35.70 | 41.00 |
| 61.00 | 50.00 | 50.00 | 45.00 | 43.00 | 47.50 | 53.60 | 54.50 | 38.00 | 35.50 | 39.50 |
| 60.00 | 45.50 | 53.51 | 47.00 | 43.00 | 47.70 | 54.61 | 44.50 | 36.00 | 36.40 | 40.00 |
| 63.00 | 43.00 | 55.00 | 48.00 | 44.00 | 49.00 | 55.00 | 44.50 | 36.80 | 36.50 | 40.00 |
| 61.50 | 46.99 | 53.50 | 48.50 | 44.00 | 50.00 | 56.00 | 46.00 | 39.50 | 37.50 | 39.00 |
| 61.00 | 48.00 | 54.00 | 50.50 | 46.00 | 49.50 | 57.20 | 45.00 | 38.00 | 38.00 | 39.00 |
| 59.50 | 50.00 | 53.00 | 50.50 | 44.50 | 49.00 | 56.10 | 45.50 | 39.50 | 38.50 | 36.50 |
| 59.00 | 49.00 | 54.00 | 48.00 | 45.50 | 50.50 | 57.00 | 44.00 | 39.00 | 40.00 | 36.50 |
| 60.50 | 45.00 | 52.60 | 45.00 | 47.50 | 51.50 | 56.50 | 41.90 | 38.00 | 40.00 | 36.00 |
| 59.80 | 45.00 | 51.00 | 41.00 | 48.00 | 51.99 | 57.00 | 41.00 | 36.00 | 42.00 | 34.80 |
| 60.00 | 43.50 | 49.80 | 38.00 | 47.99 | 52.00 | 58.50 | 40.01 | 33.00 | 43.80 | 33.00 |
| 60.00 | 41.00 | 48.00 | 43.20 | 48.00 | 51.00 | 57.50 | 39.00 | 32.50 | 43.00 | 33.00 |
| 61.00 | 38.50 | 45.50 | 48.00 | 48.00 | 52.00 | 57.00 | 39.00 | 32.50 | 44.00 | 30.01 |
| 60.00 | 38.11 | 47.50 | 45.00 | 47.40 | 54.99 | 56.00 | 38.00 | 33.00 | 44.50 | 33.00 |
| 60.00 | 44.98 | 46.00 | 44.00 | 46.50 | 56.10 | 58.00 | 38.00 | 34.50 | 45.50 | 33.00 |
| 60.51 | 43.00 | 50.00 | 44.70 | 45.50 | 57.99 | 57.50 | 37.50 | 34.40 | 44.00 | 32.00 |
| 61.28 | 43.49 | 48.50 | 46.50 | 46.00 | 57.29 | 57.00 | 36.00 | 33.21 | 42.00 | 32.50 |
Results:
| Sample | 219 |
| Mean | -0.177% |
| Median | 0.000% |
| Mode | -0.161% |
| Maximum | 18.027% |
| Minimum | -18.349% |
| St. Dev. | 4.220% |
| Skewness | 0.190 |
| Exc. Kurt. | 3.017 |
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The amount of data for returns its equal to the amount of data for quotations -1.
All the returns has out in percentual format.
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| Ranges | <= Inf. | < Sup.* | Freq. | Range mean | Range St. Dev. | | Norm. | Freq. N. | Accum. |
| 1 | -18.349% | -15.924% | 1 | -18.349% | 0.000% | | 1 | 0.188 | 0.188 |
| 2 | -15.924% | -13.499% | 0 | 0.000% | 0.000% | | 2 | 0.000 | 0.188 |
| 3 | -13.499% | -11.074% | 0 | 0.000% | 0.000% | | 3 | 0.000 | 0.188 |
| 4 | -11.074% | -8.649% | 5 | -9.382% | 0.607% | | 4 | 0.941 | 1.130 |
| 5 | -8.649% | -6.223% | 7 | -7.311% | 1.011% | | 5 | 1.318 | 2.448 |
| 6 | -6.223% | -3.798% | 19 | -5.108% | 0.577% | | 6 | 3.578 | 6.025 |
| 7 | -3.798% | -1.373% | 47 | -2.476% | 0.663% | | 7 | 8.850 | 14.875 |
| 8 | -1.373% | 1.052% | 59 | -0.207% | 0.625% | | 8 | 11.109 | 25.985 |
| 9 | 1.052% | 3.477% | 51 | 1.992% | 0.659% | | 9 | 9.603 | 35.588 |
| 10 | 3.477% | 5.902% | 16 | 4.494% | 0.626% | | 10 | 3.013 | 38.601 |
| 11 | 5.902% | 8.327% | 7 | 7.038% | 0.582% | | 11 | 1.318 | 39.919 |
| 12 | 8.327% | 10.752% | 4 | 9.441% | 0.578% | | 12 | 0.753 | 40.672 |
| 13 | 10.752% | 13.177% | 1 | 11.111% | 0.000% | | 13 | 0.188 | 40.860 |
| 14 | 13.177% | 15.602% | 1 | 13.684% | 0.000% | | 14 | 0.188 | 41.048 |
| 15 | 15.602% | 18.027% | 1 | 18.027% | 0.000% | | 15 | 0.188 | 41.237 |
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*To the last range <= Sup.
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Histogram if quotations returns informed.
Histograma= Histogram ; Ocorrências= Occurrences ; Faixas= Ranges
DP/Faixas= St. Dev./Ranges ; Freq.= Frequency ; DP Fx= St. Dev. Fx
Histograma Normalizado= Normalized histogram ; Freq. Normalizadas= Normalized Frequencies
Faixas= Ranges ; Freq. N.= N. Freq. ; Acum=Accumulated
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