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of the Central Limit Theorem

Figure 6.5 shows the distribution of the 50 data points for GDP in table 6.1. The range is quite broad, from $118 to $978 per year per person, and the shape of the distribution is multimodal.

The actual mean of the data in table 6.1—that is, the parameter we want to estimate—is $533.28. There are 2,118,760 samples of size 5 that can be taken from 50 eleTable 6.1 Per Capita Gross Domestic Product (PCGDP) in U.S. Dollars for the 50 Poorest Countries in the World, 2007

Country

PCGDP

Country

PCGDP

Burundi

118

Burkina Faso

483

DR-Congo

151

Mali

554

Zimbabwe

159

Tajikistan

555

Liberia

195

Comoros

556

Ethiopia

201

Cambodia

598

Guinea-Bissau

211

Haiti

612

Malawi

257

Benin

618

Eritrea

271

N. Korea

618

Niger

289

Ghana

647

Somalia

291

Chad

692

Sierra Leone

330

Kyrgyzstan

704

Afghanistan

345

Uzbekistan

704

Rwanda

354

Laos

711

Mozambique

362

Kiribati

762

Tanzania

368

Kenya

786

Gambia

377

Lesotho

797

Madagascar

377

Viet Nam

815

Myanmar

379

Mauritania

874

Togo

386

Senegal

908

Timor-Leste

393

Sao Tome and Principe

912

Central African Rep.

394

Papua New Guinea

953

Uganda

403

Yemen

967

Nepal

419

Zambia

974

Bangladesh

428

India

976

Guinea

452

Solomon Islands

978

SOURCE: United Nations, Dept. of Economic and Social Affairs, Economic and Social Development. http:// unstats.un.org/unsd/demographic/products/socind/inc-eco.htm.

Table 6.2 All Samples of Two from Five Elements

Sample

Mean

Cumulative mean

Uzbekistan and Senegal

(704

+

908)/2 =

806.0

806.0

Uzbekistan and Guinea

(704

+

452)/2 =

578.0

1,384.0

Uzbekistan and Rwanda

(704

+

354)/2 =

529.0

1,913.0

Uzbekistan and Liberia

(704

+

195)/2 =

449.5

2,362.5

Senegal and Guinea

(908

+

452)/2 =

680.0

3,042.5

Senegal and Rwanda

(908

+

354)/2 =

631.0

3,673.5

Senegal and Liberia

(908

+

195)/2 =

551.5

4,225.0

Guinea and Rwanda

(452

+

354)/2 =

403.0

4,628.0

Guinea and Liberia

(452

+

195)/2 =

323.5

4,951.5

Liberia and Rwanda

(195

+

354)/2 =

274.5

5,226.0

x =

5,226/10 =

522.6

ments. Table 6.3 shows the means from 10 samples of five countries chosen at random from the data in table 6.1.

Even in this small set of 10 samples, the mean is $504.72—quite close to the actual mean of $533.28. Figure 6.6 (left) shows the distribution of these samples. It has the look of a normal distribution straining to happen. Figure 6.6 (right) shows 20 samples of five from the 50 countries in table 6.1. The strain toward the normal curve is unmistakable and the mean of those 20 samples is $505.18.

The problem is that in real research, we don’t get to take 10 or 20 samples. We have

FIGURE 6.4.

Five cases and the distribution of samples of size 2 from those cases.

FIGURE 6.5.

The distribution of the 50 data points for GDP in table 6.1.

Table 6.3 10 Means from Samples of Size 5 Taken from the 50 Elements in Table 6.1

522.60

652.80

434.40

461.20

586.20

489.20

468.20

458.60

465.00

509.00

Mean = 504.72 Standard Deviation = 67.51

to make do with one. The first sample of five elements that I took had a mean of $522.60—pretty close to the actual mean of $533.28. But it’s very clear from table 6.3 that any one sample of five elements from table 6.1 could be off by a lot. They range, after all, from $434.40 to $652.80. That’s a very big spread, when the real average we’re trying to estimate is $533.28. Still, as you can see from figure 6.6, as we add samples, the mean of the samples gets closer and closer to the parameter we’re trying to estimate and the distribution of the means of the samples looks more and more like the normal distribution.

We are much closer to answering the question: How big does a sample have to be?

 
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