The Relationship between the Population and Prisoner Population of Countries around the World

Mathematics Studies Standard Level Coursework17th January 2016

Ruhi Rajani 001482-0017

North London Collegiate School

Mrs Buffham

Word Count: 2051

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In this controlled assessment I will investigate whether the prisoner population of different countries is independent of the population of those countries. I will use a selection of countries that have been randomly selected using the Excel number generator, by assigning a random number to all of the countries presented by the World Bank, and using this number to choose the countries that were generated. I will do this to ensure that any form of bias is avoided, in compliance with the basis that random sampling is 'a sampling method in which all members of a group (population or universe) have an equal and independent chance of being selected'1. I will use a sample size of 64, due to the fact that this is a large enough sample size to come to a conclusion of whether the populations of countries and their prisons are connected, however this size is not too large that it would take an extremely long time to analyse the findings that this data presents. I decided to investigate this subject due to an interest in the legal system, and I wanted to find out if the populations of different countries played a large role on the number of people that were in prison on those countries or not.  In order to carry out this investigation, I used data from the World Bank2 for the populations of each country. I then used the International Centre for Prison Studies3 to collect the data regarding the prison populations of each country. I chose these institutions as they are well-respected organisations, and therefore are likely to have provided reliable data. I decided to use data from the year 2014, as there was data available for both population and prisoner population for that year. This is not as reliable as collecting data from other years, but this was enough data for the scope of this investigation. In order to measure the relationship between these two variables, the data had to be from the same year, in order for the results of the analysis to be accurate.

I will carry out simple and complex tests to discover the mean and standard deviation of the data, as well as calculating what percentage of the population in each country are prisoners, data which I will display on a graph. These tests will help to determine if there is a relationship between the populations of countries and the number of prisoners in each country. In order to determine independence of these two factors, I will conduct a chi-squared test.

My hypothesis is that the prisoner population of countries is not independent of the population of countries, due to the logic that if there are more people in a country, there are more people to commit crimes. However, I am, factoring an awareness that in less developed countries, for example, other causes such as overpopulation can also lead to crime because of a need to obtain food and money in order to cope with the conditions of that country.

1 http://www.businessdictionary.com/definition/random-sampling.html2 http://data.worldbank.org/indicator/SP.POP.TOTL3 http://www.prisonstudies.org/highest-to-lowest/prison-population-total?field_region_taxonomy_tid=All&=Apply

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Table 1. Raw Data Sample Test 1.Country Population

(x)Prisoner Population ( y )

Percentage of Population who are Prisoners

China 1,364,270,000

1,657,812 0.12%

India 1,267,401,849

411,992 0.03%

USA 318,857,056 2,217,000 0.70%Indonesia 254,454,778 161,692 0.06%Brazil 206,077,898 607,731 0.29%Pakistan 185,132,926 80,169 0.04%Nigeria 178,516,904 57,121 0.03%Bangladesh 158,512,570 69,719 0.04%Russian Federation

143,819,569 649,500 0.45%

Japan 127,131,800 61,794 0.05%Mexico 125,385,833 255,138 0.20%Philippines 99,138,690 120,076 0.12%Ethiopia 96,958,732 93,044 0.10%Vietnam 90,730,000 142,636 0.16%Egypt 89,579,670 62,000 0.07%Germany 80,889,505 63,628 0.08%Iran 78,143,644 225,624 0.29%Turkey 75,837,020 165,033 0.22%D. Rep. Congo 74,877,030 22,000 0.03%

Thailand 67,725,979 314,303 0.46%France 66,201,365 66,864 0.10%UK 64,510,376 95,605 0.15%Italy 61,336,387 52,144 0.09%South Africa 54,001,953 159,241 0.29%Myanmar 53,437,159 60,000 0.11%Colombia 47,791,393 120,905 0.25%Spain 46,404,602 64,835 0.14%Ukraine 45,362,900 71,325 0.16%Argentina 42,980,026 64,288 0.15%Algeria 38,934,334 60,220 0.15%Poland 37,995,529 75,691 0.20%Morocco 33,921,203 76,000 0.22%Peru 30,973,148 74,486 0.24%Madagascar 23,571,962 18,719 0.08%Australia 23,490,736 35,804 0.15%Romania 19,910,995 29,312 0.15%Niger 18,534,802 7,116 0.04%Netherlands 16,854,183 12,638 0.07%Malawi 16,829,144 12,156 0.07%

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Ecuador 15,982,551 25,902 0.16%Zambia 15,021,002 17,160 0.11%Rwanda 12,100,049 55,618 0.46%Cuba 11,258,597 57,337 0.51%Greece 10,957,740 11,988 0.11%Haiti 10,461,409 10,266 0.10%Portugal 10,397,393 14,293 0.14%Hungary 9,861,673 17,716 0.18%Sweden 9,689,555 5,779 0.06%Austria 8,534,492 8,241 0.10%Israel 8,215,300 18,658 0.23%Switzerland 8,190,229 6,923 0.08%Serbia 7,129,428 10,031 0.14%Libya 6,253,452 6,187 0.10%Denmark 5,639,565 3,481 0.06%Norway 5,136,475 3,710 0.07%Ireland 4,612,719 3,791 0.08%New Zealand 4,509,700 8,641 0.19%Croatia 4,236,400 3,853 0.09%Jamaica 2,721,252 4,050 0.15%Iceland 327,589 147 0.04%Barbados 286,066 884 0.31%Greenland 56,295 116 0.21%Monaco 38,066 23 0.06%Tuvalu 9,894 11 0.11%

In order to calculate these percentages I used the following equation: prisoner population

population×100

For example:

prisoner populationof China

population of China×100= 1,657,812

1,364,270,000×100=0.12151 = 0.12%

These percentages display that there is a relationship between the prisoner population and the population of countries. There are however certain outliers which are significant, with the USA having a higher percentage (0.7%) than most countries, and India having a lower percentage (0.03%) than most countries, both of these countries having two of the biggest populations out of all of the countries. There are several countries, such as Rwanda, Cuba and the Russian Federation which have a significantly higher prisoner population than the countries that have a similar sized smaller population to them, but the majority of these countries follow the trend of there being a general positive correlation.

I decided to present this information on a scatter graph to display the correlation as one that is more significant.

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This graph presents a problem in the claim that there is a correlation, due to the range of populations meaning that the data is not spread out, and highly concentrated in the bottom left corner of the graph. However, it clearly shows the outliers, with the USA having an incredibly large prisoner population and a lower population than both China and India, which are the other two major outliers. India has a much lower prisoner population than other countries which are seen to have a significantly lower population than it. These anomalies could be to do with the larger land mass of these countries, combined with their large population. It could also potentially be due to the rigorousness of the legal systems in those countries, perhaps with the USA having a more rigorous system than other countries.

This table displays the process used to find the mean and standard deviation of the data.

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0 200000000 400000000 600000000 800000000 1000000000 1200000000 1400000000 16000000000

500000

1000000

1500000

2000000

2500000

Prison Population

Population

Pris

oner

Pop

ulat

ion

100000000 150000000

Table 2. Population (x)Prisoner

Population ( y )

(x−x ) (x−x )2 ( y− y) ( y− y)2

China 1364270000 1657812 1270549523 1614296089879370000 1518903 2307066750600India 1267401849 411992 1173681372 1377527962502990000 273083 74574401694USA 318857056 2217000 225136579 50686479112361500 2078091 4318462788744Indonesia 254454778 161692 160734301 25835515452660300 22783 519071497Brazil 206077898 607731 112357421 12624190008126000 468822 219794199540Pakistan 185132926 80169 91412449 8356235795041290 -58740 3450371079Nigeria 178516904 57121 84796427 7190433997517780 -81788 6689253941Bangladesh 158512570 69719 64792093 4198015288998860 -69190 4787236640Russian Federation

143819569 649500 50099092 2509918998871710 510591 260703312885

Japan 127131800 61794 33411323 1116316491036980 -77115 5946701536Mexico 125385833 255138 31665356 1002694757742690 116229 13509213130Philippines 99138690 120076 5418213 29357029912220 -18833 354676592Ethiopia 96958732 93044 3238255 10486294129484 -45865 2103585325Vietnam 90730000 142636 -2990477 8942953902410 3727 13891577Egypt 89579670 62000 -4140807 17146284293452 -76909 5914972650Germany 80889505 63628 -12830972 164633847677366 -75281 5667207788Iran 78143644 225624 -15576833 242637732637977 86715 7519515614Turkey 75837020 165033 75837020 5751253602480400 165033 27235891089D. Rep. Congo

74877030 22000 -18843447 355075502496959 -116909 13667681400

Thailand 67725979 314303 -25994498 675713936832269 175394 30763104566France 66201365 66864 -27519112 757301536448183 -72045 5190461762UK 64510376 95605 -29210101 853230012296805 -43304 1875224237Italy 61336387 52144 -32384090 1048729298284140 -86765 7528140822South Africa 54001953 159241 -39718524 1577561164874230 20332 413395942Myanmar 53437159 60000 -40283318 1622745725454220 -78909 6226608088Colombia 47791393 120905 -45929084 2109480775737750 -18004 324138952Spain 46404602 64835 -47315875 2238792046237700 -74074 5486936643Ukraine 45362900 71325 -48357577 2338455272956190 -67584 4567578048Argentina 42980026 64288 -50740451 2574593388296710 -74621 5568272654Algeria 38934334 60220 -54786143 3001521487073320 -78689 6191936590Poland 37995529 75691 -55724948 3105269852240960 -63218 3996497744Morocco 33921203 76000 -59799274 3575953195220530 -62909 3957524588Peru 30973148 74486 -62747329 3937227322125340 -64423 4150304810Madagascar 23571962 18719 -70148515 4920814185203060 -120190 14445602297Australia 23490736 35804 -70229741 4932216549457910 -103105 10630612027Romania 19910995 29312 -73809482 5447839663093430 -109597 12011471585Niger 18534802 7116 -75185675 5652885755749810 -131793 17369357782Netherlands 16854183 12638 -76866294 5908427184521370 -126271 15944329927Malawi 16829144 12156 -76891333 5912277121753990 -126753 16066287360Ecuador 15982551 25902 -77737926 6043185170362510 -113007 12770550266Zambia 15021002 17160 -78699475 6193607397247290 -121749 14822784759

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Rwanda 12100049 55618 -81620428 6661894300061480 -83291 6937367255Cuba 11258597 57337 -82461880 6799961686634540 -81572 6653968242Greece 10957740 11988 -82762737 6849670669353530 -126921 16108904544Haiti 10461409 10266 -83259068 6932072438052620 -128643 16548985268Portugal 10397393 14293 -83323084 6942736361121060 -124616 15529112408Hungary 9861673 17716 -83858804 7032299042378050 -121193 14687709163Sweden 9689555 5779 -84030922 7061195886307650 -133130 17723559457Austria 8534492 8241 -85185985 7256652075027030 -130668 17074089474Israel 8215300 18658 -85505177 7311135328537810 -120251 14460269180Switzerland 8190229 6923 -85530248 7315423357688170 -131986 17420267075Serbia 7129428 10031 -86591049 7498009802098010 -128878 16609502637Libya 6253452 6187 -87467025 7650480497884100 -132722 17615091956Denmark 5639565 3481 -88080912 7758247094534610 -135428 18340705095Norway 5136475 3710 -88584002 7847125446323250 -135199 18278731576Ireland 4612719 3791 -89107758 7940192571986590 -135118 18256835922New Zealand 4509700 8641 -89210777 7958562769185610 -130268 16969715186Croatia 4236400 3853 -89484077 8007400072894830 -135056 18240085152Jamaica 2721252 4050 -90999225 8280858987569060 -134859 18186911952Iceland 327589 147 -93392888 8722231566921400 -138762 19254853617Barbados 286066 884 -93434411 8729989196874650 -138025 19050861805Greenland 56295 116 -93664182 8772979027780200 -138793 19263457813Monaco 38066 23 -93682411 8776394168831400 -138886 19289281934Tuvalu 9894 11 -93710583 8781673404269810 -138898 19292615339Total: 64 0 3357806963765090000 138909 7860074726825

Standard Deviation:

σ=√∑ ¿¿¿¿

σ=√ 335780696376509000064

=¿52465733808829500

From this standard deviation, we can infer that the data is widely spread out, showing that our data does not fall close to the mean values. This spread of the data means that there is not a significant correlation between the population of countries and the prisoner population.

Following on from the results in the statistical test for significance, I will conduct a χ2test.

H0=prisoner population is independent of population

H1=prisoner population is not independent of population

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Table 3- Observed data, collected and collated from tables 1 and 2  Prison

Population   

Population 1-60,000 60,000+  Totals0-60,000,000 31 10 41

60,000,001-120,000,000 2 10 12

120,000,001+ 1 10 11

 Totals 34 30 64

The data was arranged this way in order to allow the expected values to be large enough to be able to calculate the χ2 value.

In order to calculate the expected values for the first row, for example, I used the equation:

Expected Value= RowTotal ×ColumnTotalTotal

Expected Value=41×3464

=21.78125

The Expected data calculated was:

Table 4.  Prison

Population   

Population 1-60,000 60,000+  Totals0-60,000,000 21.78125 19.21875 41

60,000,001-120,000,000 6.375 5.625 12

120,000,001+ 5.84375 5.15625 11

 Totals 34 30 64

The data from both of these tables was then combined, and used to calculate theχ2 value, using the equation:

χ2=∑ ¿¿¿Table 5.

Observed Values Expected Values Observed-Expected (Observed-Expected)2

(Observed-Expected)2/Expected

31 21.78125 9.21875 84.9854 3.90177

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Observed Values Expected Values Observed-Expected (Observed-Expected)2

(Observed-Expected)2/Expected

10 19.21875 -9.21875 -84.9854 4.422

2 6.375 -4.375 -19.1406 3.00245

10 5.625 4.375 19.1406 3.40277

1 5.84375 -4.84375 -23.4619 4.01487

10 5.15625 4.84375 23.4619 4.55019

23.2941

Method 1:

Degrees of freedom= (number of rows-1) x (number of columns-1)= (3-1) x (2-1)= 2 x 1= 2

Critical value for 2= 5.991 at 5% level4

23.2941>5.991

χ2>critical value

Therefore, we reject H0 hypothesis. This suggests that prisoner population is not independent of population.

Method 2:

From GDC: p-value is 0.000009.

0.000009<0.05

Therefore we reject H0 hypothesis. This again suggests that prisoner population is not independent of population.

Conclusion:

From the test conducted on the data of the prisoner population and the populations of different countries, we can draw the conclusion that prisoner population is dependent on population, meaning that there is a link between the two variables. The chi squared test enabled me to reject the null hypothesis in two instances, ensuring that the results were accurate. However, the test conducted on the mean and standard deviation presents clear anomalies, and a positive correlation is not found, due to the large spread of the data. It can therefore be concluded that although the population and prisoner population are not independent of one another, there is no clear correlation between the two sets of data. Bibliography:

1. http://www.businessdictionary.com/definition/random-sampling.html2. http://data.worldbank.org/indicator/SP.POP.TOTL3. http://www.prisonstudies.org/highest-to-lowest/prison-population-total?field_region_taxonomy_tid=All&=Apply4. http://www.itl.nist.gov/div898/handbook/eda/section3/eda3674.htm

4 http://www.itl.nist.gov/div898/handbook/eda/section3/eda3674.htm9