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The Sample Variance – Explanation & Examples

The definition of the sample variance is:

“The sample variance is the average of the squared differences from the mean found in a sample.”

In this topic, we will discuss the sample variance from the following aspects:

  • What is the sample variance?
  • How to find the sample variance?
  • Sample variance formula.
  • The role of the sample variance.
  • Practice questions.
  • Answer key.

What is the sample variance?

The sample variance is the average of the squared differences from the mean found in a sample.

The sample variance measures the spread of a numerical characteristic of your sample.

A large variance indicates that your sample numbers are far from the mean and far from each other.

A small variance, on the other hand, indicates the opposite.

A zero variance indicates that all values within your sample are identical.

The variance can be zero or a positive number. Still, it cannot be negative because it is mathematically impossible to have a negative value resulting from a square.

For example, if you have two sets of 3 numbers (1,2,3) and (1,2,10). You see that the second set is more spread (more varied) than the first set.

You can see that from the following dot plot.

We see that the blue dots (second group) are more spread out than the red dots (first group).

If we calculate the first group variance, it is 1, while the variance for the second group is 24.3. Therefore, the second group is more spread (more varied) than the first group.

How to find the sample variance?

We will go through several examples, from simple to more complex ones.

– Example 1

What is the variance of the numbers, 1,2,3?

1. Add up all of the numbers:

1+2+3 = 6.

2. Count the numbers of items in your sample. In this sample, there are 3 items.

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

The sample mean = 6/3 = 2.

4. In a table, subtract the mean from each value of your sample.

value

value-mean

1

-1

2

0

3

1

You have a table of 2 columns, one for the data values and the other column for subtracting the mean (2) from each value.

4. Add another column for the squared differences you found in Step 4.

value

value-mean

squared difference

1

-1

1

2

0

0

3

1

1

6. Add up all of the squared differences you found in Step 5.

1+0+1 = 2.

7. Divide the number you get in step 6 by sample size-1 to get the variance. We have 3 numbers, so the sample size is 3.

The variance = 2/(3-1) = 1.

– Example 2

What is the variance of the numbers, 1,2,10?

1. Add up all of the numbers:

1+2+10 = 13.

2. Count the numbers of items in your sample. In this sample, there are 3 items.

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

The sample mean = 13/3 = 4.33.

4. In a table, subtract the mean from each value of your sample.

value

value-mean

1

-3.33

2

-2.33

10

5.67

You have a table of 2 columns, one for the data values and the other column for subtracting the mean (4.33) from each value.

5. Add another column for the squared differences you found in Step 4.

value

value-mean

squared difference

1

-3.33

11.09

2

-2.33

5.43

10

5.67

32.15

6. Add up all of the squared differences you found in Step 5.

11.09 + 5.43 + 32.15 = 48.67.

7. Divide the number you get in step 6 by sample size-1 to get the variance. We have 3 numbers, so the sample size is 3.

The variance = 48.67/(3-1) = 24.335.

– Example 3

The following is the age (in years) of 25 individuals sampled from a certain population. What is the variance of this sample?

individual

age

1

26

2

48

3

67

4

39

5

25

6

25

7

36

8

44

9

44

10

47

11

53

12

52

13

52

14

51

15

52

16

40

17

77

18

44

19

40

20

45

21

48

22

49

23

19

24

54

25

82

1. Add up all of the numbers:

26+ 48+ 67+ 39+ 25+ 25+ 36+ 44+ 44+ 47+ 53+ 52+ 52+ 51+ 52+ 40+ 77+ 44+ 40+ 45+ 48+ 49+ 19+ 54+ 82 = 1159.

2. Count the numbers of items in your sample. In this sample, there are 25 items or 25 individuals.

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

The sample mean = 1159/25 = 46.36 years.

4. In a table, subtract the mean from each value of your sample.

individual

age

age-mean

1

26

-20.36

2

48

1.64

3

67

20.64

4

39

-7.36

5

25

-21.36

6

25

-21.36

7

36

-10.36

8

44

-2.36

9

44

-2.36

10

47

0.64

11

53

6.64

12

52

5.64

13

52

5.64

14

51

4.64

15

52

5.64

16

40

-6.36

17

77

30.64

18

44

-2.36

19

40

-6.36

20

45

-1.36

21

48

1.64

22

49

2.64

23

19

-27.36

24

54

7.64

25

82

35.64

There is one column for the ages and another column for subtracting the mean (46.36) from each value.

5. Add another column for the squared differences you found in Step 4.

individual

age

age-mean

squared difference

1

26

-20.36

414.53

2

48

1.64

2.69

3

67

20.64

426.01

4

39

-7.36

54.17

5

25

-21.36

456.25

6

25

-21.36

456.25

7

36

-10.36

107.33

8

44

-2.36

5.57

9

44

-2.36

5.57

10

47

0.64

0.41

11

53

6.64

44.09

12

52

5.64

31.81

13

52

5.64

31.81

14

51

4.64

21.53

15

52

5.64

31.81

16

40

-6.36

40.45

17

77

30.64

938.81

18

44

-2.36

5.57

19

40

-6.36

40.45

20

45

-1.36

1.85

21

48

1.64

2.69

22

49

2.64

6.97

23

19

-27.36

748.57

24

54

7.64

58.37

25

82

35.64

1270.21

6. Add up all of the squared differences you found in Step 5.

414.53+ 2.69+ 426.01+ 54.17+ 456.25+ 456.25+ 107.33+ 5.57+ 5.57+ 0.41+ 44.09+ 31.81+ 31.81+ 21.53+ 31.81+ 40.45+ 938.81+ 5.57+ 40.45+ 1.85+ 2.69+ 6.97+ 748.57+ 58.37+ 1270.21 = 5203.77.

7. Divide the number you get in step 6 by sample size-1 to get the variance. We have 25 numbers so the sample size is 25.

The variance = 5203.77/(25-1) = 216.82 years^2.

Note that the sample variance has the squared unit of the original data (years^2) due to the presence of squared difference in its calculation.

– Example 4

The following is the score (in points) of 10 students in an easy exam. What is the variance of this sample?

student

score

1

100

2

100

3

100

4

100

5

100

6

100

7

100

8

100

9

100

10

100

All students have 100 points on this exam.

1. Add up all of the numbers:

Sum = 1000.

2. Count the numbers of items in your sample. In this sample, there are 10 items or students.

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

The sample mean = 1000/10 = 100.

4. In a table, subtract the mean from each value of your sample.

student

score

score-mean

1

100

0

2

100

0

3

100

0

4

100

0

5

100

0

6

100

0

7

100

0

8

100

0

9

100

0

10

100

0

5. Add another column for the squared differences you found in Step 4.

student

score

score-mean

squared difference

1

100

0

0

2

100

0

0

3

100

0

0

4

100

0

0

5

100

0

0

6

100

0

0

7

100

0

0

8

100

0

0

9

100

0

0

10

100

0

0

6. Add up all of the squared differences you found in Step 5.

Sum = 0.

7. Divide the number you get in step 6 by sample size-1 to get the variance. We have 10 numbers, so the sample size is 10.

The variance = 0/(10-1) = 0 points^2.

The variance can be zero if all our sample values are identical.

– Example 5

The following table shows the daily closing prices (in US dollars or USD) of Facebook (FB) and Google (GOOG) stocks in some days of 2013. Which stock has a more variable closing stock price?

Note that we compare the two stocks from the same sector (communication services) and for the same period.

date

FB

GOOG

2013-01-02

28.00

723.2512

2013-01-03

27.77

723.6713

2013-01-04

28.76

737.9713

2013-01-07

29.42

734.7513

2013-01-08

29.06

733.3012

2013-01-09

30.59

738.1212

2013-01-10

31.30

741.4813

2013-01-11

31.72

739.9913

2013-01-14

30.95

723.2512

2013-01-15

30.10

724.9313

2013-01-16

29.85

715.1912

2013-01-17

30.14

711.3212

2013-01-18

29.66

704.5112

2013-01-22

30.73

702.8712

2013-01-23

30.82

741.5013

2013-01-24

31.08

754.2113

2013-01-25

31.54

753.6713

2013-01-28

32.47

750.7313

2013-01-29

30.79

753.6813

2013-01-30

31.24

753.8313

2013-01-31

30.98

755.6913

2013-02-01

29.73

775.6013

2013-02-04

28.11

759.0213

2013-02-05

28.64

765.7413

2013-02-06

29.05

770.1713

2013-02-07

28.65

773.9513

2013-02-08

28.55

785.3714

2013-02-11

28.26

782.4213

2013-02-12

27.37

780.7013

2013-02-13

27.91

782.8613

2013-02-14

28.50

787.8214

2013-02-15

28.32

792.8913

2013-02-19

28.93

806.8514

2013-02-20

28.46

792.4613

2013-02-21

27.28

795.5313

2013-02-22

27.13

799.7114

2013-02-25

27.27

790.7714

2013-02-26

27.39

790.1313

2013-02-27

26.87

799.7813

2013-02-28

27.25

801.2014

2013-03-01

27.78

806.1914

2013-03-04

27.72

821.5014

2013-03-05

27.52

838.6014

2013-03-06

27.45

831.3814

2013-03-07

28.58

832.6014

2013-03-08

27.96

831.5214

2013-03-11

28.14

834.8214

2013-03-12

27.83

827.6114

2013-03-13

27.08

825.3114

2013-03-14

27.04

821.5414

We will calculate the variance for each stock then compare between them.

The variance of Facebook stock closing price is calculated as follows:

1. Add up all of the numbers:

28.00+ 27.77+ 28.76+ 29.42+ 29.06+ 30.59+ 31.30+ 31.72+ 30.95+ 30.10+ 29.85+ 30.14+ 29.66+ 30.73+ 30.82+ 31.08+ 31.54+ 32.47+ 30.79+ 31.24+ 30.98+ 29.73+ 28.11+ 28.64+ 29.05+ 28.65+ 28.55+ 28.26+ 27.37+ 27.91+ 28.50+ 28.32+ 28.93+ 28.46+ 27.28+ 27.13+ 27.27+ 27.39+ 26.87+ 27.25+ 27.78+ 27.72+ 27.52+ 27.45+ 28.58+ 27.96+ 28.14+ 27.83+ 27.08+ 27.04 = 1447.74.

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

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

The sample mean = 1447.74/50 = 28.9548 USD.

4. In a table, subtract the mean from each value of your sample.

FB

stock-mean

28.00

-0.9548

27.77

-1.1848

28.76

-0.1948

29.42

0.4652

29.06

0.1052

30.59

1.6352

31.30

2.3452

31.72

2.7652

30.95

1.9952

30.10

1.1452

29.85

0.8952

30.14

1.1852

29.66

0.7052

30.73

1.7752

30.82

1.8652

31.08

2.1252

31.54

2.5852

32.47

3.5152

30.79

1.8352

31.24

2.2852

30.98

2.0252

29.73

0.7752

28.11

-0.8448

28.64

-0.3148

29.05

0.0952

28.65

-0.3048

28.55

-0.4048

28.26

-0.6948

27.37

-1.5848

27.91

-1.0448

28.50

-0.4548

28.32

-0.6348

28.93

-0.0248

28.46

-0.4948

27.28

-1.6748

27.13

-1.8248

27.27

-1.6848

27.39

-1.5648

26.87

-2.0848

27.25

-1.7048

27.78

-1.1748

27.72

-1.2348

27.52

-1.4348

27.45

-1.5048

28.58

-0.3748

27.96

-0.9948

28.14

-0.8148

27.83

-1.1248

27.08

-1.8748

27.04

-1.9148

There is one column for the stock prices and another column for subtracting the mean (28.9548) from each value.

5. Add another column for the squared differences you found in Step 4.

FB

stock-mean

squared difference

28.00

-0.9548

0.91

27.77

-1.1848

1.40

28.76

-0.1948

0.04

29.42

0.4652

0.22

29.06

0.1052

0.01

30.59

1.6352

2.67

31.30

2.3452

5.50

31.72

2.7652

7.65

30.95

1.9952

3.98

30.10

1.1452

1.31

29.85

0.8952

0.80

30.14

1.1852

1.40

29.66

0.7052

0.50

30.73

1.7752

3.15

30.82

1.8652

3.48

31.08

2.1252

4.52

31.54

2.5852

6.68

32.47

3.5152

12.36

30.79

1.8352

3.37

31.24

2.2852

5.22

30.98

2.0252

4.10

29.73

0.7752

0.60

28.11

-0.8448

0.71

28.64

-0.3148

0.10

29.05

0.0952

0.01

28.65

-0.3048

0.09

28.55

-0.4048

0.16

28.26

-0.6948

0.48

27.37

-1.5848

2.51

27.91

-1.0448

1.09

28.50

-0.4548

0.21

28.32

-0.6348

0.40

28.93

-0.0248

0.00

28.46

-0.4948

0.24

27.28

-1.6748

2.80

27.13

-1.8248

3.33

27.27

-1.6848

2.84

27.39

-1.5648

2.45

26.87

-2.0848

4.35

27.25

-1.7048

2.91

27.78

-1.1748

1.38

27.72

-1.2348

1.52

27.52

-1.4348

2.06

27.45

-1.5048

2.26

28.58

-0.3748

0.14

27.96

-0.9948

0.99

28.14

-0.8148

0.66

27.83

-1.1248

1.27

27.08

-1.8748

3.51

27.04

-1.9148

3.67

6. Add up all of the squared differences you found in Step 5.

0.91+ 1.40+ 0.04+ 0.22+ 0.01+ 2.67+ 5.50+ 7.65+ 3.98+ 1.31+ 0.80+ 1.40+ 0.50+ 3.15+ 3.48+ 4.52+ 6.68+ 12.36+ 3.37+ 5.22+ 4.10+ 0.60+ 0.71+ 0.10+ 0.01+ 0.09+ 0.16+ 0.48+ 2.51+ 1.09+ 0.21+ 0.40+ 0.00+ 0.24+ 2.80+ 3.33+ 2.84+ 2.45+ 4.35+ 2.91+ 1.38+ 1.52+ 2.06+ 2.26+ 0.14+ 0.99+ 0.66+ 1.27+ 3.51+ 3.67 = 112.01.

7. Divide the number you get in step 6 by sample size-1 to get the variance. We have 50 numbers so the sample size is 50.

8. The variance of Facebook stock closing price = 112.01/(50-1) = 2.29 USD^2.

The variance of Google stock closing price is calculated as follows:

1. Add up all of the numbers:

723.2512+ 723.6713+ 737.9713+ 734.7513+ 733.3012+ 738.1212+ 741.4813+ 739.9913+ 723.2512+ 724.9313+ 715.1912+ 711.3212+ 704.5112+ 702.8712+ 741.5013+ 754.2113+ 753.6713+ 750.7313+ 753.6813+ 753.8313+ 755.6913+ 775.6013+ 759.0213+ 765.7413+ 770.1713+ 773.9513+ 785.3714+ 782.4213+ 780.7013+ 782.8613+ 787.8214+ 792.8913+ 806.8514+ 792.4613+ 795.5313+ 799.7114+ 790.7714+ 790.1313+ 799.7813+ 801.2014+ 806.1914+ 821.5014+ 838.6014+ 831.3814+ 832.6014+ 831.5214+ 834.8214+ 827.6114+ 825.3114+ 821.5414 = 38622.02.

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

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

The sample mean = 38622.02/50 = 772.4404 USD.

4. In a table, subtract the mean from each value of your sample.

GOOG

stock-mean

723.2512

-49.1892

723.6713

-48.7691

737.9713

-34.4691

734.7513

-37.6891

733.3012

-39.1392

738.1212

-34.3192

741.4813

-30.9591

739.9913

-32.4491

723.2512

-49.1892

724.9313

-47.5091

715.1912

-57.2492

711.3212

-61.1192

704.5112

-67.9292

702.8712

-69.5692

741.5013

-30.9391

754.2113

-18.2291

753.6713

-18.7691

750.7313

-21.7091

753.6813

-18.7591

753.8313

-18.6091

755.6913

-16.7491

775.6013

3.1609

759.0213

-13.4191

765.7413

-6.6991

770.1713

-2.2691

773.9513

1.5109

785.3714

12.9310

782.4213

9.9809

780.7013

8.2609

782.8613

10.4209

787.8214

15.3810

792.8913

20.4509

806.8514

34.4110

792.4613

20.0209

795.5313

23.0909

799.7114

27.2710

790.7714

18.3310

790.1313

17.6909

799.7813

27.3409

801.2014

28.7610

806.1914

33.7510

821.5014

49.0610

838.6014

66.1610

831.3814

58.9410

832.6014

60.1610

831.5214

59.0810

834.8214

62.3810

827.6114

55.1710

825.3114

52.8710

821.5414

49.1010

There is one column for the stock prices and another column for subtracting the mean (772.4404) from each value.

5. Add another column for the squared differences you found in Step 4.

GOOG

stock-mean

squared difference

723.2512

-49.1892

2419.58

723.6713

-48.7691

2378.43

737.9713

-34.4691

1188.12

734.7513

-37.6891

1420.47

733.3012

-39.1392

1531.88

738.1212

-34.3192

1177.81

741.4813

-30.9591

958.47

739.9913

-32.4491

1052.94

723.2512

-49.1892

2419.58

724.9313

-47.5091

2257.11

715.1912

-57.2492

3277.47

711.3212

-61.1192

3735.56

704.5112

-67.9292

4614.38

702.8712

-69.5692

4839.87

741.5013

-30.9391

957.23

754.2113

-18.2291

332.30

753.6713

-18.7691

352.28

750.7313

-21.7091

471.29

753.6813

-18.7591

351.90

753.8313

-18.6091

346.30

755.6913

-16.7491

280.53

775.6013

3.1609

9.99

759.0213

-13.4191

180.07

765.7413

-6.6991

44.88

770.1713

-2.2691

5.15

773.9513

1.5109

2.28

785.3714

12.9310

167.21

782.4213

9.9809

99.62

780.7013

8.2609

68.24

782.8613

10.4209

108.60

787.8214

15.3810

236.58

792.8913

20.4509

418.24

806.8514

34.4110

1184.12

792.4613

20.0209

400.84

795.5313

23.0909

533.19

799.7114

27.2710

743.71

790.7714

18.3310

336.03

790.1313

17.6909

312.97

799.7813

27.3409

747.52

801.2014

28.7610

827.20

806.1914

33.7510

1139.13

821.5014

49.0610

2406.98

838.6014

66.1610

4377.28

831.3814

58.9410

3474.04

832.6014

60.1610

3619.35

831.5214

59.0810

3490.56

834.8214

62.3810

3891.39

827.6114

55.1710

3043.84

825.3114

52.8710

2795.34

821.5414

49.1010

2410.91

6. Add up all of the squared differences you found in Step 5.

2419.58+ 2378.43+ 1188.12+ 1420.47+ 1531.88+ 1177.81+ 958.47+ 1052.94+ 2419.58+ 2257.11+ 3277.47+ 3735.56+ 4614.38+ 4839.87+ 957.23+ 332.30+ 352.28+ 471.29+ 351.90+ 346.30+ 280.53+ 9.99+ 180.07+ 44.88+ 5.15+ 2.28+ 167.21+ 99.62+ 68.24+ 108.60+ 236.58+ 418.24+ 1184.12+ 400.84+ 533.19+ 743.71+ 336.03+ 312.97+ 747.52+ 827.20+ 1139.13+ 2406.98+ 4377.28+ 3474.04+ 3619.35+ 3490.56+ 3891.39+ 3043.84+ 2795.34+ 2410.91 = 73438.76.

7. Divide the number you get in step 6 by sample size-1 to get the variance. We have 50 numbers, so the sample size is 50.

The Google stock closing price variance = 73438.76/(50-1) = 1498.75 USD^2, while the variance of the Facebook stock closing price is 2.29 USD^2.

The Google stock closing price is more variable. We can see that if we plot the data as a dot plot.

In the first plot, when the x-axis is common, we see that the Facebook prices occupy a small space compared to Google prices.

In the second plot, when the x-axis values are set according to each stock’s values, we see that Facebook prices range from 27 to 32, while Google prices range from 700 to about 850.

Sample variance formula

The sample variance formula is:

s^2=(∑_(i=1)^n▒( x_i-¯x )^2)/(n-1)

Where s^2 is the sample variance.

¯x is the sample mean.

n is the sample size.

The term:

∑_(i=1)^n▒( x_i-¯x )^2

means sum the squared difference between every element of our sample (from x_1 to x_n) and the sample mean ¯x.

Our sample element is denoted as x with a subscript to indicate its position in our sample.

In the example of stock prices for Facebook, we have 50 prices. The first price (28) is denoted as x_1, the second price (27.77) is denoted as x_2, the third price (28.76) is denoted as x_3.

The last price (27.04) is denoted as x_50 or x_n because n = 50 in this case.

We used this formula in the above examples, where we summed the squared difference between every element of our sample and the sample mean, then divided by the sample size-1 or n-1.

We divide by n-1 when calculating the sample variance (and not by n as any average) to make the sample variance a good estimator of the true population variance.

If you have population data, you will divide by N (where N is the population size) to get the variance.

– Example

We have a population of more than 20,000 individuals. From the census data, the true population variance for the age was 298.84 years^2.

We take a random sample of 50 individuals from this data. The sum of squared differences from the mean was 12112.08.

If we divide by 50 (sample size), the variance will be 242.24, while if we divide by 49 (sample size-1), the variance will be 247.19.

Dividing by n-1 prevents the sample variance from underestimating the true population variance.

The role of the sample variance

The sample variance is a summary statistic that can be used to deduce the spread of the population from which the sample was randomly selected.

In the above example about Google and Facebook stock prices, although we have only a sample of 50 days, we can conclude (with some level of certainty) Google stock is more variable (riskier) than Facebook stock.

Variance is important in an investment where we can use it (as a measure of spread or variability) as a measure of risk.

We see in the above example that although Google stock has a higher closing price, it is more variable and so more risky to invest in.

Another example is when the product produced from some machines is with high variance in the industrial machines. It indicates that these machines need adjustment.

Disadvantages of variance as a measure of spread:

  1. It is affected by outliers. These are the numbers that are far from the mean. Squaring the differences between these numbers and the mean can skew the variance.
  2. Not easily interpreted because the variance has the squared unit of the data.

We use the variance to take the square root of its value, which indicates the standard deviation of the data set. Thus, the standard deviation has the same unit as the original data, so it is more easily interpreted.

Practice questions

1. The following table is the daily closing prices (in USD) of two stocks from the financial sector, JP Morgan Chase (JPM) and Citigroup (C), for some days in 2011. Which stock has a more variable closing stock price?

Date

JP Morgan

Citigroup

2011-06-01

41.76

39.65

2011-06-02

41.61

40.01

2011-06-03

41.57

39.85

2011-06-06

40.53

38.07

2011-06-07

40.72

37.58

2011-06-08

40.39

36.81

2011-06-09

40.98

37.77

2011-06-10

41.05

37.92

2011-06-13

41.67

39.17

2011-06-14

41.61

38.78

2011-06-15

40.68

38.00

2011-06-16

40.36

37.63

2011-06-17

40.80

38.30

2011-06-20

40.48

38.16

2011-06-21

40.91

39.31

2011-06-22

40.69

39.51

2011-06-23

40.07

39.41

2011-06-24

39.49

39.59

2011-06-27

39.88

39.99

2011-06-28

39.54

40.15

2011-06-29

40.45

41.50

2011-06-30

40.94

41.64

2011-07-01

41.58

42.88

2011-07-05

41.03

42.57

2011-07-06

40.56

42.01

2011-07-07

41.32

42.63

2011-07-08

40.74

42.03

2011-07-11

39.43

39.79

2011-07-12

39.39

39.07

2011-07-13

39.62

39.47

2. The following is a table of the compressive strengths for 25 concrete samples (in pound per square inch or psi) produced from 3 different machines. Which machine is more precise in its production?

Note more precise means less variable.

machine_1

machine_2

machine_3

12.55

26.86

66.70

37.68

53.30

28.47

76.80

23.25

21.86

25.12

20.08

28.80

12.45

15.34

26.91

36.80

37.44

64.90

48.40

15.69

11.85

59.80

23.69

31.87

48.15

37.27

15.09

39.23

44.61

52.42

40.86

64.90

77.30

42.33

10.22

48.67

46.23

25.51

29.65

19.35

29.79

37.68

32.04

11.47

50.46

35.17

23.79

24.28

31.35

28.63

39.30

6.28

30.12

33.36

40.06

8.06

28.63

40.60

33.80

35.75

33.72

32.25

35.10

46.64

55.64

6.47

29.89

71.30

37.42

16.50

67.11

12.64

30.45

40.06

51.26

3. The following is a table for the variance in weights of diamonds produced from 4 different machines and a dot plot for the individual weight values.

machine

variance

machine_1

0.2275022

machine_2

0.3267417

machine_3

0.1516739

machine_4

0.1873904

We see that machine_3 has the least variance. Knowing that, which dots are most likely produced from machine_3?

4. The following is the variance for different stocks closing prices (from the same sector). Which stock is safer to invest in?

symbol2

variance

stock_1

30820.2059

stock_2

971.7809

stock_3

31816.9763

stock_4

26161.1889

5. The following dot plot is for the daily Ozone measurements in New York, May to September 1973. Which month is the most variable in Ozone measurements, and which month is the least variable?

Answer key

1. We will calculate the variance for each stock then compare between them.

The variance of JP Morgan Chase stock closing price is calculated as follows:

  • Add up all of the numbers:

Sum = 1219.85.

  • Count the numbers of items in your sample. In this sample, there are 30 items.
  • Divide the number you found in step 1 by the number you found in step 2.

The sample mean = 1219.85/30 = 40.66167.

  • Subtract the mean from each value of your sample and square the difference.

JP Morgan

stock-mean

squared difference

41.76

1.0983

1.21

41.61

0.9483

0.90

41.57

0.9083

0.83

40.53

-0.1317

0.02

40.72

0.0583

0.00

40.39

-0.2717

0.07

40.98

0.3183

0.10

41.05

0.3883

0.15

41.67

1.0083

1.02

41.61

0.9483

0.90

40.68

0.0183

0.00

40.36

-0.3017

0.09

40.80

0.1383

0.02

40.48

-0.1817

0.03

40.91

0.2483

0.06

40.69

0.0283

0.00

40.07

-0.5917

0.35

39.49

-1.1717

1.37

39.88

-0.7817

0.61

39.54

-1.1217

1.26

40.45

-0.2117

0.04

40.94

0.2783

0.08

41.58

0.9183

0.84

41.03

0.3683

0.14

40.56

-0.1017

0.01

41.32

0.6583

0.43

40.74

0.0783

0.01

39.43

-1.2317

1.52

39.39

-1.2717

1.62

39.62

-1.0417

1.09

  • Add up all of the squared differences you found in Step 4.

Sum = 14.77.

  • Divide the number you get in step 5 by sample size-1 to get the variance. We have 30 numbers, so the sample size is 30.

The variance of JPM stock closing price = 14.77/(30-1) = 0.51 USD^2.

The variance of Citigroup stock closing price is calculated as follows:

  • Add up all of the numbers:

Sum = 1189.25.

  • Count the numbers of items in your sample. In this sample, there are 30 items.
  • Divide the number you found in step 1 by the number you found in step 2.

Тhe sample mean = 1189.25/30 = 39.64167.

  • Subtract the mean from each value of your sample and square the difference.

Citigroup

stock-mean

squared difference

39.65

0.0083

0.00

40.01

0.3683

0.14

39.85

0.2083

0.04

38.07

-1.5717

2.47

37.58

-2.0617

4.25

36.81

-2.8317

8.02

37.77

-1.8717

3.50

37.92

-1.7217

2.96

39.17

-0.4717

0.22

38.78

-0.8617

0.74

38.00

-1.6417

2.70

37.63

-2.0117

4.05

38.30

-1.3417

1.80

38.16

-1.4817

2.20

39.31

-0.3317

0.11

39.51

-0.1317

0.02

39.41

-0.2317

0.05

39.59

-0.0517

0.00

39.99

0.3483

0.12

40.15

0.5083

0.26

41.50

1.8583

3.45

41.64

1.9983

3.99

42.88

3.2383

10.49

42.57

2.9283

8.57

42.01

2.3683

5.61

42.63

2.9883

8.93

42.03

2.3883

5.70

39.79

0.1483

0.02

39.07

-0.5717

0.33

39.47

-0.1717

0.03

  • Add up all of the squared differences you found in step 4.

Sum = 80.77.

  • Divide the number you get in step 5 by sample size-1 to get the variance. We have 30 numbers, so the sample size is 30.

Citigroup stock closing price variance = 80.77/(30-1) = 2.79 USD^2, while the variance of JP Morgan Chase stock closing price is only 0.51 USD^2.

The Citigroup stock closing price is more variable. We can see that if we plot the data as a dot plot.

When the x-axis is common, we see that the Citigroup prices are more scattered than JP Morgan prices.

2. We will calculate the variance for each machine then compare them.

The variance of machine_1 is calculated as follows:

  •  Add up all of the numbers:

Sum = 888.45.

  • Count the numbers of items in your sample. In this sample, there are 25 items.
  • Divide the number you found in step 1 by the number you found in step 2.

The sample mean = 888.45/25 = 35.538.

  • Subtract the mean from each value of your sample and square the difference.

machine_1

strength-mean

squared difference

12.55

-22.988

528.45

37.68

2.142

4.59

76.80

41.262

1702.55

25.12

-10.418

108.53

12.45

-23.088

533.06

36.80

1.262

1.59

48.40

12.862

165.43

59.80

24.262

588.64

48.15

12.612

159.06

39.23

3.692

13.63

40.86

5.322

28.32

42.33

6.792

46.13

46.23

10.692

114.32

19.35

-16.188

262.05

32.04

-3.498

12.24

35.17

-0.368

0.14

31.35

-4.188

17.54

6.28

-29.258

856.03

40.06

4.522

20.45

40.60

5.062

25.62

33.72

-1.818

3.31

46.64

11.102

123.25

29.89

-5.648

31.90

16.50

-19.038

362.45

30.45

-5.088

25.89

  • Add up all of the squared differences you found in Step 4.

Sum = 5735.17.

  • Divide the number you get in step 5 by sample size-1 to get the variance. We have 25 numbers, so the sample size is 25.

The variance of machine_1 = 5735.17/(25-1) = 238.965 psi^2.

With similar calculations, the variance of machine_2 = 315.6805 psi^2, and the variance for machine_3 = 310.7079 psi^2.

The machine_1 is more precise or less variable in the compressive strength of concrete produced.

3. Blue dots because they are more compact than other dot groups.

4. Stock_2 because it has the least variance.

5. The most variable month is 8 or August and the least variable month is 6 or June.

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