The Population Mean – Explanation & Examples

The definition of the population mean is:

“The population mean is the mean or average found in a population.”

In this topic, we will discuss the population mean from the following aspects:

  • What is the population mean?
  • How to find the population mean?
  • The population mean formula.
  • The role of the population mean.
  • Practice questions.
  • Answer key.

What is the population mean?

The population mean is the mean value of a numerical characteristic of the population. The population is the whole group of items we want to study. These items can be individuals, things, animals, plants, etc.

For example, the whole individuals living in the U.S., the whole chairs produced from a certain factory, the whole tigers living in rain forests in Indonesia, and the whole orange trees in Egypt.

These different populations’ numerical characteristics can be weights for individuals, leg lengths for chairs, tail lengths for tigers, and the heights for orange trees.

However, collecting information from the population may not be possible in many cases due to the great resources it needs.

For example, if we want to study the heights of American males. We can survey every American male and get his height. This is population data.

Alternatively, we can select 200 American males and measure their heights. This is sample data.

If we calculate the mean of the population data, its symbol is the Greek letter μ and pronounced “mu.”

How to find the population mean?

We have two cases:

  1. We have population data and so calculate the population mean from it.
  2. We have sample data and use the sample mean to construct an interval that most likely contains the population mean.

– Examples of population data

– Example 1

The following is the murder rate (per 100,000 population) for the 50 states of the U.S. in 1976. What is the mean of the murder rate?

We have information about all states of the U.S. so this is population data.

Note

This dataset may be considered as a sample or as a population.

It is a sample if we want to study the murder rate of the U.S. states in the 1970s, or it is a population if data is from 1970-1980 because this is a 1-year sample from these 10 years.

It is a population for the murder rate of U.S. states in 1976.

state

murder rate

Alabama

15.1

Alaska

11.3

Arizona

7.8

Arkansas

10.1

California

10.3

Colorado

6.8

Connecticut

3.1

Delaware

6.2

Florida

10.7

Georgia

13.9

Hawaii

6.2

Idaho

5.3

Illinois

10.3

Indiana

7.1

Iowa

2.3

Kansas

4.5

Kentucky

10.6

Louisiana

13.2

Maine

2.7

Maryland

8.5

Massachusetts

3.3

Michigan

11.1

Minnesota

2.3

Mississippi

12.5

Missouri

9.3

Montana

5.0

Nebraska

2.9

Nevada

11.5

New Hampshire

3.3

New Jersey

5.2

New Mexico

9.7

New York

10.9

North Carolina

11.1

North Dakota

1.4

Ohio

7.4

Oklahoma

6.4

Oregon

4.2

Pennsylvania

6.1

Rhode Island

2.4

South Carolina

11.6

South Dakota

1.7

Tennessee

11.0

Texas

12.2

Utah

4.5

Vermont

5.5

Virginia

9.5

Washington

4.3

West Virginia

6.7

Wisconsin

3.0

Wyoming

6.9

1. Add up all of the numbers:

15.1+ 11.3+ 7.8+ 10.1+ 10.3+ 6.8+ 3.1+ 6.2+ 10.7+ 13.9+ 6.2+ 5.3+ 10.3+ 7.1+ 2.3+ 4.5+ 10.6+ 13.2+ 2.7+ 8.5+ 3.3+ 11.1+ 2.3+ 12.5+ 9.3+ 5.0+ 2.9+ 11.5+ 3.3+ 5.2+ 9.7+ 10.9+ 11.1+ 1.4+ 7.4+ 6.4+ 4.2+ 6.1+ 2.4+ 11.6+ 1.7+ 11.0+ 12.2+ 4.5+ 5.5+ 9.5+ 4.3+ 6.7+ 3.0+ 6.9 = 368.9.

2. Count the numbers of items in your population. In this population, there are 50 items or 50 states.

3. Divide the number you found in Step 1 by the number you found in Step 2.

The population mean = 368.9/50 = 7.378.

Note that the population mean has the same unit as the original data. So 7.378 is the mean murder rate per 100,000 population.

– Example 2

The following are the weights (in grams) for 71 chickens on a certain farm. What is the mean?

We have the weights of all chickens on the farm, so this is population data.

chicken number

weight

1

179

2

160

3

136

4

227

5

217

6

168

7

108

8

124

9

143

10

140

11

309

12

229

13

181

14

141

15

260

16

203

17

148

18

169

19

213

20

257

21

244

22

271

23

243

24

230

25

248

26

327

27

329

28

250

29

193

30

271

31

316

32

267

33

199

34

171

35

158

36

248

37

423

38

340

39

392

40

339

41

341

42

226

43

320

44

295

45

334

46

322

47

297

48

318

49

325

50

257

51

303

52

315

53

380

54

153

55

263

56

242

57

206

58

344

59

258

60

368

61

390

62

379

63

260

64

404

65

318

66

352

67

359

68

216

69

222

70

283

71

332

1. Add up all of the numbers:

179+ 160+ 136+ 227+ 217+ 168+ 108+ 124+ 143+ 140+ 309+ 229+ 181+ 141+ 260+ 203+ 148+ 169+ 213+ 257+ 244+ 271+ 243+ 230+ 248+ 327+ 329+ 250+ 193+ 271+ 316+ 267+ 199+ 171+ 158+ 248+ 423+ 340+ 392+ 339+ 341+ 226+ 320+ 295+ 334+ 322+ 297+ 318+ 325+ 257+ 303+ 315+ 380+ 153+ 263+ 242+ 206+ 344+ 258+ 368+ 390+ 379+ 260+ 404+ 318+ 352+ 359+ 216+ 222+ 283+ 332 = 18553.

2. Count the numbers of items in your population. In this population, there are 71 items or chickens.
3. Divide the number you found in Step 1 by the number you found in Step 2.

The population mean = 18553/71 = 261.3 grams.

– Example 3

The following is the trunk circumference (in mm) for 35 orange trees on a certain farm. What is the mean?

We have the trunk circumferences of all trees on the farm, so this is population data.

tree number

circumference

1

30

2

58

3

87

4

115

5

120

6

142

7

145

8

33

9

69

10

111

11

156

12

172

13

203

14

203

15

30

16

51

17

75

18

108

19

115

20

139

21

140

22

32

23

62

24

112

25

167

26

179

27

209

28

214

29

30

30

49

31

81

32

125

33

142

34

174

35

177

1. Add up all of the numbers:

30+ 58+ 87+ 115+ 120+ 142+ 145+ 33+ 69+ 111+ 156+ 172+ 203+ 203+ 30+ 51+ 75+ 108+ 115+ 139+ 140+ 32+ 62+ 112+ 167+ 179+ 209+ 214+ 30+ 49+ 81+ 125+ 142+ 174+ 177 = 4055.

2. Count the numbers of items in your population. In this population, there are 35 items or trees.

3. Divide the number you found in Step 1 by the number you found in Step 2.

The population mean = 4055/35 = 115.8571 mm.

– Examples of sample data

For samples with a size greater than 30, the interval that, most likely, contains the population mean is calculated by:

¯x±1.96Xs/√n

Where:

¯x is the calculated sample mean.

s is the standard deviation of the sample. It is a measure of the data spread.

n is the sample size.

This interval (called 95% confidence interval) gives us a range of possible values for the unknown population mean from which the sample was taken.

– Example 4

The following is the age (in years) of 50 randomly selected individuals from a certain population. If you know that this sample’s standard deviation is 18.65, construct a 95% confidence interval for the true population mean.

89 61 74 85 46 60 41 18 37 30 44 37 51 53 74 38 56 48 52 62 33 56 38 30 43 32 74 27 49 53 40 27 42 60 88 22 59 43 69 75 28 47 35 62 65 31 22 31 26 83.

1. Add up all of the numbers in your sample:

89+ 61+ 74+ 85+ 46+ 60+ 41+ 18+ 37+ 30+ 44+ 37+ 51+ 53+ 74+ 38+ 56+ 48+ 52+ 62+ 33+ 56+ 38+ 30+ 43+ 32+ 74+ 27+ 49+ 53+ 40+ 27+ 42+ 60+ 88+ 22+ 59+ 43+ 69+ 75+ 28+ 47+ 35+ 62+ 65+ 31+ 22+ 31+ 26+ 83 = 2446.

2. Count the numbers of items in your sample. In this sample, there are 50 items or persons.

3. Divide the number you found in Step 1 by the number you found in Step 2.

The sample mean = 2446/50 = 48.92 years.

4. The 95% confidence interval is:

¯x±1.96Xs/√n

¯x-1.96Xs/√n to ¯x+1.96Xs/√n

48.92-1.96X18.65/√50 to 48.92+1.96X18.65/√50 or 43.75 to 54.1.

It means that the true population mean age can be as small as 43.75 years and as large as 54.1 years.

Owing to the presence of the √n term in the formula for an interval calculation, the sample size affects the interval width. Larger sample sizes lead to smaller interval widths (or a more precise estimate of the population mean).

Suppose that you have a 100 sample size and you obtain the same sample mean and standard deviation; the 95% confidence interval will be:

48.92-1.96X18.65/√100 to 48.92+1.96X18.65/√100 or 45.26 to 52.6.

Suppose that you have a 500 sample size and you obtain the same sample mean and standard deviation; the 95% confidence interval will be:

48.92-1.96X18.65/√500 to 48.92+1.96X18.65/√500 or 47.29 to 50.55.

With increasing the sample size, you have more values about the true population mean.

We can see that in the following figure.

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