Cross Mark and Laws of Demand and Supply

Different symbols have different meanings although the meaning could be contextual and vary. Here, I want to discuss a cross mark symbol, which I guess most of us have been using since childhood. Cross mark is usually used to indicate that an answer is wrong or an option of a response at which a cross mark is place is not applicable to the person cross marking.

A cross mark gave me another meaning also. If I place a cross mark on a graph it gives me a mining of laws of demand and supply although the cross mark may not exactly represent the demand and supply curves. The market determines the price of a commodity or a service based on demand and supply of a good or service. Price remains at the Y-axis and quantity demanded or supplied at the X-axis of a chart.

The law of demand states that demand will be less at a higher price of a good or service in the context that other things remaining constant. The demand curve is symbolized by a descending line from left to right of a cross mark.

The law of supply states that supply will be more as price increases in the context that other things remaining constant, The supply curve is symbolized by an ascending line from left to right of a cross mark.

These two laws interact at an equilibrium price, a meeting point of demand and supply curves in a cross mark, at which the seller can sell the quantity s/he wants to sell and the buyer can buy the quantity s/he wants to buy. Thus, a cross mark symbolizes the laws of demand and supply.

Tick Mark and Resilience

Different symbols have different meanings although the meaning could be contextual and vary. Here, I want to discuss a tick mark symbol, which I guess most have been using since childhood. Tick mark is usually used to indicate that an answer is correct or an option of a response at which a tick mark is place is applicable to the person tick marking.

A tick mark gave me another meaning also. If I place a tick mark on a graph it gives me a meaning of resilience. The term resilience has been a buzz word in academia and development, community resilience, household resilience, resilience to climate change so and so forth. An entity is considered resilient if it bounces or has the capacity to bounce back after hitting by shock or stress. Once an entity is hit by a shock or stress the current level of capacity of an entity could come down from the left tip of a tick mark to the lowest point of it. But, as time elapses and the context improves, gradually the capacity of an entity could also increase from the lowest point of a tick mark to its highest point. It could increase further as there is no hard and fast limitation of the tip of a tick mark. The resilient capacity could vary from an entity to another. Some may bounce back quickly and others may take a long time. However, in the natural process, an entity could face a shock or stress and could come back to its previous level or better situation as a tick mark. Thus, I wish everyone to be like a tick mark.

Exploring Quantitative Data

Exploring data is a kind of data health check to see whether there are outliers that cause abnormal distribution. Parametric inferential statistics are designed to fit the normal distribution. Several summary functions and graphical displays are used to explore data. I exemplify these measures from the daily database on COVID-19 incidence worldwide as of April 29, 2020, which I downloaded from the European Centre for Disease Prevention and Control website. The dataset has 109 daily records. This article needs to read along with my previous article entitled ‘Describing Quantitative Data’.

Five Number Summary

Minimum value, first quartile (Q1) or 25^th percentile, median or 50^th percentile, third quartile (Q3) or 75^th percentile and maximum value are five summary numbers used to explore data. In a normal distribution the median is at the center of the distribution. Besides, the difference between the minimum value and median is equal to the difference between the median and maximum value. Likewise, the difference between the minimum value and Q1 is equal to the difference between Q3 and maximum value as shown by black and red arrows in figure 1.

Figure 1: Five Number Summary in Normal Distribution

In positively skewed distribution distance between minimum value and median is smaller than the distance between median and maximum value as shown by arrows in figure 2. Likewise, the distance between the minimum value and Q1 is smaller than the distance between Q3 and maximum value.

Figure 2: Five Number Summary in Positively Skewed Distribution

In negatively skewed distribution the distance between minimum value and median is greater than the distance between median and maximum value as shown by black and red arrows in figure 3. Likewise, the distance between the minimum value and Q1 is greater than the distance between Q3 and maximum value.

Figure 3: Five Number Summary in Negatively Skewed Distribution

In the given dataset, the minimum value is one and the maximum value is 101,728. Q1, median, and Q3 are respectively 1761.5, 3907, and 66403.5. The difference between minimum value and median is equal to 3906 which is far smaller than the difference between the median and maximum value equal to 97,821.  This is consistent with figure 2 indicating that data is positively skewed.

Normality Checks

Box plot displays the five number summaries as shown in figure 4. The thick black line inside the box symbolizes the median. The lower and upper hinges or boundaries of the box symbolize Q1 and Q3, respectively. Whiskers below and above the hinges symbolize the minimum and maximum values. Box plot shows the outliers and extreme values. Outliers are the values below or above the value calculated as one and half times the difference between Q1 and Q3. The given dataset does not have statistically outlying values as there are no indications on the box plot. However, there are five extreme values above Q3 from 86,046 to the maximum number of persons infected in a day. Likewise, five extreme values below Q1 from 17 to the minimum number of infected cases. Those values have led to skewed distribution.

The stem and leaf plot shows the shape of distribution showing positive skew statistic in which the values cluster below 10,000 cases per day having 62 data points. Values from 50,000 to below 90,000 cases are also clustered as shown in figure 5. This shows that the distribution is positively skewed.

Figure 5: Stem and leaf plot

A normal Q-Q plot displays the straight line that represents expected values for normal distribution. The observed incidence values deviate distinctly from that line indicating positively skewed distribution as shown in figure 6.

Figure 6: Normal Q-Q Plot

In a nutshell, exploring data summarizes distribution, identifies outliers, and checks normality. These procedures confirm that the distribution of the given dataset is positively skewed.

Descriptive Measures of Quantitative Data

Descriptive statistics measure central tendency, dispersion, and distribution. I exemplify these measures from the daily database on COVID-19 incidence worldwide as of April 29, 2020, which I downloaded from the European Centre for Disease Prevention and Control website. The dataset has 109 daily records.

Measure of Central Tendency

Mean incidence per day is 28,016.6 persons. Median or second quartile is located at 3,907 persons and mode is 17 persons infected in a day. Mean, median, and mode are equal for normal distribution. Both median and mode are less than mean indicating that the dataset is distorted from normality.

Measure of Dispersion

Range of 101,727 is very wide between a maximum of 101,728 persons to one person infected on a single day worldwide. The standard deviation is calculated at 34,079.4 persons. For the normal distribution, 95 percent of the data values fall within two standard deviations from the mean. In this dataset, 95 percent of data values fall within plus 96,175.4 to minus 40,142.2. 

Measures of Distribution

Skewness value is positive 0.73.  A normal distribution has zero skewness. The standard error of skewness is 0.23. The positive value indicates that the distribution is positively skewed with a right tail. Since a skewness value is more than two times its standard error the distribution is asymmetric.

Kurtosis value is negative 1.24. A normal distribution has a kurtosis value of zero. The standard error of kurtosis is 0.45. The negative value indicates that the distribution is flatter than normal with the data values less clustered around the center of the distribution, which is also referred to as platykurtic distribution. Since a kurtosis value is more than two times its standard error the distribution is abnormal.

In a nutshell, descriptive statistics measure the central tendency, variability, and distribution of data. The given dataset is positively skewed and flatly distributed than normal. 

Frequency, Tabulation and Graphical Presentation

Frequency is a summary measure that describes both categorical and quantitative data. It counts the number of times a category or a data value is repeated. The summary data can be presented in tables and graphs. I exemplify from the database on COVID-19 incidence and human death worldwide as of April 28, 2020, which I downloaded from the European Centre for Disease Prevention and Control website.

Categorical Data

I was curious to know how many countries in different continents or regions had had deaths.  I found that 169 out of 206 countries had deaths and 37 did not deaths as shown in Table 1. Most European countries had deaths and most African countries did not have deaths.

Table 1: Countries Reporting Death Due to COVID-19

The above data can be visualized in a bar chart for better presentation as shown in Figure 1.

Quantitative Data

I was curious to know how many countries had how many numbers of death due to COVID-19 falling in a certain class interval. I found that most countries (161 out of 169) reporting death had deaths falling within a class interval of one to less than 5000.  It was followed by three countries that had deaths in the interval between 5,000 to less than 10,000. One country had deaths within the range of 55,000 to less than 60,000 (Table 2).

Table 2: Country and Death Due to COVID-19

The above data can be presented in the histogram as shown in Figure 2. Data shows that the number of countries is positively skewed.

I was curious to know how many persons died at what time interval. It was appraised that in April 2020 alone 82 percent of total 209,776 people died. It seems that the number of deaths is rapidly increasing by the month. It could be because people who suffered in initial months started to die.

Table 3: Death by Month in 2020

Above data can also be visualized in the Time series line chart in Figure 3.

In a nutshell, frequency is a process of summarizing data, which can be presented in tabular and graphical forms.

How Predictable Death by COVID-19 Incidence?

I downloaded from the European Centre for Disease Prevention and Control website the daily database constituting the number of new cases reported and death worldwide due to COVID-19. The database had 13,623 daily records from December 31, 2019 to April 28, 2020. I summed records by country and filtered the number of incidences avoiding outliers using a statistical formula.  The filtered record shows a list of 136 countries that had an incidence in 314,099 persons, of which 7,866 persons died. It shows that 2.5 percent of infected people had already died and there is a possibility of other deaths too, which the time will tell. Israel had a maximum of 15,598 incidences, and the British Virgin Islands had a minimum of 6 incidences. Algeria had a maximum of 437 deaths and 13 countries each had one death.

Then, I calculated the Pearson correlation coefficient between the number of incidence and deaths for the filtered records.  The purpose of removing outliers was because the Pearson correlation coefficient works when the datasets are close to normal distribution. The analysis shows a correlation coefficient of 0.677 which was significantly different from zero (p=0.01). This suggests that there is a moderate chance of death if one is infected with COVID-19. A regression analysis was also conducted to predict death based on incidence, which gives an R square value of 0.459 indicating that nearly half the variation in the number of deaths is explained by the model. A significant coefficient indicates that the number of incidences significantly contributes to the model. For every 1000 incidences 18 deaths are likely.

Graphics Visualizing Target Population

Graphics is an important form to visualize one data from others. This technique is useful to visualize the target population for any purpose. For example, I referred to Nepal’s population census 2011 data to get population values to visualize the number of male household heads from urban areas of Nepal belonging to the age group of 50 years and above. I used four criteria to filter and get target population data – male, age group, administrative setting (urban), and position or role in a household (household head).

An outermost blue circle shows the total population of Nepal, which is 26,494,504. It is followed by a red circle showing the male population which is 49 percent of the total population. An inner third yellow circle shows the male population aged 50 years and above, which is 15 percent of the total male population. An inner dark ash-colored circle shows the urban male population aged 50 years and above, which is 15 percent of the total males aged 50 years and above. Lastly, an innermost green circle shows the number of urban male household heads aged 50 years and above, which is 81 percent of total urban males aged 50 years and above and 1.9 percent of the total male population in Nepal.

These data are visualized in both stacked Venn and horizontal hierarchical diagrams in Figures 1 and 2 respectively.

Figure 1: Target population of urban male household heads aged 50 years and above shown in a stacked Venn diagram

Figure 2: Target population of urban male household heads aged 50 years and above shown in a horizontally hierarchical diagram

In a nutshell, visualizing filtered data in different diagrammatic forms is a cool way.