# 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.

## 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.

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?

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.