Question

for the following set of data, find the percentage of data within 2 population standard deviations of the mean, to the nearest percent.
67, 39, 42, 66, 70, 72, 70, 66, 67
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answer

Explanation:

Step1: Calculate the population mean ($\mu$)

The data set is \( 67, 39, 42, 66, 70, 72, 70, 66, 67 \). The number of data points \( N = 9 \).
\[ \mu=\frac{67 + 39 + 42 + 66 + 70 + 72 + 70 + 66 + 67}{9} \]
\[ \mu=\frac{569}{9}\approx63.22 \]

Step2: Calculate the population standard deviation ($\sigma$)

First, find the squared differences from the mean:
\((67 - 63.22)^2\approx14.29\), \((39 - 63.22)^2\approx586.61\), \((42 - 63.22)^2\approx449.29\), \((66 - 63.22)^2\approx7.73\), \((70 - 63.22)^2\approx45.97\), \((72 - 63.22)^2\approx77.09\), \((70 - 63.22)^2\approx45.97\), \((66 - 63.22)^2\approx7.73\), \((67 - 63.22)^2\approx14.29\)
Sum of squared differences:
\[ 14.29+586.61 + 449.29+7.73+45.97+77.09+45.97+7.73+14.29 = 1259.97 \]
Population variance \(\sigma^{2}=\frac{1259.97}{9}\approx139.997\)
Population standard deviation \(\sigma=\sqrt{139.997}\approx11.83\)

Step3: Find the range within \( \mu\pm2\sigma\)

Lower bound: \(63.22-2\times11.83 = 63.22 - 23.66 = 39.56\)
Upper bound: \(63.22+2\times11.83 = 63.22 + 23.66 = 86.88\)

Step4: Count data points within the range

Check each data point:
- 67: within range
- 39: \(39\lt39.56\)? No (close, but 39 < 39.56)
- 42: within range
- 66: within range
- 70: within range
- 72: within range
- 70: within range
- 66: within range
- 67: within range

Wait, correction: 39 is 39, lower bound is 39.56, so 39 is below. 42 is above 39.56. Let's recount:
Data points: 67 (yes), 39 (no), 42 (yes), 66 (yes), 70 (yes), 72 (yes), 70 (yes), 66 (yes), 67 (yes). So number of data points within range: 8 (since 39 is out, others 8 are in).

Step5: Calculate the percentage

Percentage \(=\frac{8}{9}\times100\%\approx88.89\%\approx89\%\) (Wait, wait, let's recheck the lower bound. Wait, \(\mu = 63.22\), \(\sigma\approx11.83\), \(2\sigma\approx23.66\), so \(\mu - 2\sigma=63.22 - 23.66 = 39.56\). The data point 39: 39 vs 39.56. 39 is less than 39.56, so it's outside. 42 is 42 > 39.56, so inside. Now count the data points:
1. 67: inside
2. 39: outside
3. 42: inside
4. 66: inside
5. 70: inside
6. 72: inside
7. 70: inside
8. 66: inside
9. 67: inside

So that's 8 data points inside. So percentage is \(\frac{8}{9}\times100\approx88.89\%\), which is approximately 89%. But wait, maybe my calculation of the mean was wrong. Let's recalculate the mean:

Sum of data: 67 + 39 = 106; 106 + 42 = 148; 148 + 66 = 214; 214 + 70 = 284; 284 + 72 = 356; 356 + 70 = 426; 426 + 66 = 492; 492 + 67 = 559? Wait, wait, earlier I had 569, that was a mistake! Oh no, that's the error. Let's recalculate the sum:

67 + 39 = 106

106 + 42 = 148

148 + 66 = 214

214 + 70 = 284

284 + 72 = 356

356 + 70 = 426

426 + 66 = 492

492 + 67 = 559. Oh! I added wrong before. So \(N = 9\), sum is 559. So mean \(\mu=\frac{559}{9}\approx62.11\)

Now recalculate standard deviation:

Squared differences from 62.11:

(67 - 62.11)^2 = (4.89)^2 ≈23.91

(39 - 62.11)^2 = (-23.11)^2 ≈534.07

(42 - 62.11)^2 = (-20.11)^2 ≈404.41

(66 - 62.11)^2 = (3.89)^2 ≈15.13

(70 - 62.11)^2 = (7.89)^2 ≈62.25

(72 - 62.11)^2 = (9.89)^2 ≈97.81

(70 - 62.11)^2 = (7.89)^2 ≈62.25

(66 - 62.11)^2 = (3.89)^2 ≈15.13

(67 - 62.11)^2 = (4.89)^2 ≈23.91

Sum of squared differences: 23.91 + 534.07 + 404.41 + 15.13 + 62.25 + 97.81 + 62.25 + 15.13 + 23.91 = let's add:

23.91 + 534.07 = 557.98

557.98 + 404.41 = 962.39

962.39 + 15.13 = 977.52

977.52 + 62.25 = 1039.77

1039.77 + 97.81 = 1137.58

1137.58 + 62.25 = 1199.83

1199.83 + 15.13 = 1214.96

1214.96 + 23.91 = 1238.87

Population variance \(\sigma^{2}=\frac{1238.87}{9}\approx137.65\)

Population standard deviation \(\sigma=\sqrt{137.65}\approx11.73\)

Now find \( \mu\pm2\sigma\):

Lower bound: \(62.11 - 2\times11.73 = 62.11 - 23.46 = 38.65\)

Upper bound: \(62.11 + 2\times11.73 = 62.11 + 23.46 = 85.57\)

Now check each data point:

- 67: 67 is between 38.65 and 85.57: yes
- 39: 39 > 38.65: yes (39 - 38.65 = 0.35, so inside)
- 42: yes
- 66: yes
- 70: yes
- 72: 72 < 85.57: yes
- 70: yes
- 66: yes
- 67: yes

So all 9 data points? Wait 39: 39 is greater than 38.65, so yes. Let's check:

38.65 < 39 < 85.57: yes. 42: yes. 66: yes. 70: yes. 72: yes. 70: yes. 66: yes. 67: yes. So all 9 data points are within \( \mu\pm2\sigma\)? Wait that can't be. Wait 72 is 72, upper bound is 85.57, so yes. 39 is 39, lower bound 38.65, so 39 > 38.65, so yes. So all 9 data points are within 2 standard deviations?

Wait let's recalculate the mean correctly. Data points: 67, 39, 42, 66, 70, 72, 70, 66, 67. Let's sum again:

67 + 39 = 106

106 + 42 = 148

148 + 66 = 214

214 + 70 = 284

284 + 72 = 356

356 + 70 = 426

426 + 66 = 492

492 + 67 = 559. Yes, sum is 559. Mean is 559/9 ≈62.111...

Now calculate each deviation:

67 - 62.111 ≈4.889, squared ≈23.90

39 - 62.111 ≈-23.111, squared ≈534.12

42 - 62.111 ≈-20.111, squared ≈404.45

66 - 62.111 ≈3.889, squared ≈15.12

70 - 62.111 ≈7.889, squared ≈62.24

72 - 62.111 ≈9.889, squared ≈97.80

70 - 62.111 ≈7.889, squared ≈62.24

66 - 62.111 ≈3.889, squared ≈15.12

67 - 62.111 ≈4.889, squared ≈23.90

Sum of squared deviations: 23.90 + 534.12 = 558.02; +404.45 = 962.47; +15.12 = 977.59; +62.24 = 1039.83; +97.80 = 1137.63; +62.24 = 1199.87; +15.12 = 1214.99; +23.90 = 1238.89.

Population variance: 1238.89 / 9 ≈137.654

Population standard deviation: sqrt(137.654) ≈11.732

Now 2*sigma ≈23.464

Mean - 2*sigma ≈62.111 - 23.464 ≈38.647

Mean + 2*sigma ≈62.111 + 23.464 ≈85.575

Now check each data point:

- 67: 38.647 < 67 < 85.575: yes
- 39: 38.647 < 39 < 85.575: yes (39 - 38.647 ≈0.353)
- 42: yes
- 66: yes
- 70: yes
- 72: 72 < 85.575: yes
- 70: yes
- 66: yes
- 67: yes

So all 9 data points are within 2 standard deviations of the mean. Wait, that's because 39 is just above the lower bound (38.647), so it's included. So the number of data points within the range is 9. Therefore, the percentage is (9/9)*100% = 100%? But that seems off. Wait maybe I made a mistake in the data set. Let's list the data again: 67, 39, 42, 66, 70, 72, 70, 66, 67. Let's sort them: 39, 42, 66, 66, 67, 67, 70, 70, 72.

Now, mean is ~62.11, sigma ~11.73. So 2 sigma below mean is ~38.65, 2 sigma above is ~85.58. All data points are between 39 and 72, which is within 38.65 and 85.58. So yes, all 9 data points are within 2 standard deviations. Therefore, the percentage is 100%? Wait but that seems unusual. Wait maybe the initial sum was wrong. Let's add again: 67 + 39 = 106; +42 = 148; +66 = 214; +70 = 284; +72 = 356; +70 = 426; +66 = 492; +67 = 559. Yes, sum is 559. 559/9 = 62.111...

Wait, maybe the problem is a population, so we use population standard deviation. Let's check with a calculator approach. Alternatively, maybe the empirical rule, but the empirical rule is for normal distributions, but here we calculate directly.

Wait, let's use a calculator for the data set: 39, 42, 66, 66, 67, 67, 70, 70, 72.

Mean: (39 + 42 + 66 + 66 + 67 + 67 + 70 + 70 + 72)/9 = (39+42=81; 66+66=132; 67+67=134; 70+70=140; 72. So 81+132=213; +134=347; +140=487; +72=559. 559/9≈62.11.

Population standard deviation:

Using the formula \(\sigma=\sqrt{\frac{\sum (x_i - \mu)^2}{N}}\)

We calculated \(\sum (x_i - \mu)^2 = 1238.89\), so \(\sigma=\sqrt{1238.89/9}=\sqrt{137.654}\approx11.73\)

Now, 2*sigma≈23.46. So the interval is (62.11 - 23.46, 62.11 + 23.46) = (38.65, 85.57). Now, all data points:

Answer:

Step1: Calculate the population mean ($\mu$)

The data set is \( 67, 39, 42, 66, 70, 72, 70, 66, 67 \). The number of data points \( N = 9 \).

$$ \mu=\frac{67 + 39 + 42 + 66 + 70 + 72 + 70 + 66 + 67}{9} $$

$$ \mu=\frac{569}{9}\approx63.22 $$

Step2: Calculate the population standard deviation ($\sigma$)

First, find the squared differences from the mean:
\((67 - 63.22)^2\approx14.29\), \((39 - 63.22)^2\approx586.61\), \((42 - 63.22)^2\approx449.29\), \((66 - 63.22)^2\approx7.73\), \((70 - 63.22)^2\approx45.97\), \((72 - 63.22)^2\approx77.09\), \((70 - 63.22)^2\approx45.97\), \((66 - 63.22)^2\approx7.73\), \((67 - 63.22)^2\approx14.29\)
Sum of squared differences:

$$ 14.29+586.61 + 449.29+7.73+45.97+77.09+45.97+7.73+14.29 = 1259.97 $$

Population variance \(\sigma^{2}=\frac{1259.97}{9}\approx139.997\)
Population standard deviation \(\sigma=\sqrt{139.997}\approx11.83\)

Step3: Find the range within \( \mu\pm2\sigma\)

Lower bound: \(63.22-2\times11.83 = 63.22 - 23.66 = 39.56\)
Upper bound: \(63.22+2\times11.83 = 63.22 + 23.66 = 86.88\)

Step4: Count data points within the range

Check each data point:

• 67: within range
• 39: \(39\lt39.56\)? No (close, but 39 < 39.56)
• 42: within range
• 66: within range
• 70: within range
• 72: within range
• 70: within range
• 66: within range
• 67: within range

Wait, correction: 39 is 39, lower bound is 39.56, so 39 is below. 42 is above 39.56. Let's recount:
Data points: 67 (yes), 39 (no), 42 (yes), 66 (yes), 70 (yes), 72 (yes), 70 (yes), 66 (yes), 67 (yes). So number of data points within range: 8 (since 39 is out, others 8 are in).

Step5: Calculate the percentage

Percentage \(=\frac{8}{9}\times100\%\approx88.89\%\approx89\%\) (Wait, wait, let's recheck the lower bound. Wait, \(\mu = 63.22\), \(\sigma\approx11.83\), \(2\sigma\approx23.66\), so \(\mu - 2\sigma=63.22 - 23.66 = 39.56\). The data point 39: 39 vs 39.56. 39 is less than 39.56, so it's outside. 42 is 42 > 39.56, so inside. Now count the data points:

1. 67: inside
2. 39: outside
3. 42: inside
4. 66: inside
5. 70: inside
6. 72: inside
7. 70: inside
8. 66: inside
9. 67: inside

So that's 8 data points inside. So percentage is \(\frac{8}{9}\times100\approx88.89\%\), which is approximately 89%. But wait, maybe my calculation of the mean was wrong. Let's recalculate the mean:

Sum of data: 67 + 39 = 106; 106 + 42 = 148; 148 + 66 = 214; 214 + 70 = 284; 284 + 72 = 356; 356 + 70 = 426; 426 + 66 = 492; 492 + 67 = 559? Wait, wait, earlier I had 569, that was a mistake! Oh no, that's the error. Let's recalculate the sum:

67 + 39 = 106

106 + 42 = 148

148 + 66 = 214

214 + 70 = 284

284 + 72 = 356

356 + 70 = 426

426 + 66 = 492

492 + 67 = 559. Oh! I added wrong before. So \(N = 9\), sum is 559. So mean \(\mu=\frac{559}{9}\approx62.11\)

Now recalculate standard deviation:

Squared differences from 62.11:

(67 - 62.11)^2 = (4.89)^2 ≈23.91

(39 - 62.11)^2 = (-23.11)^2 ≈534.07

(42 - 62.11)^2 = (-20.11)^2 ≈404.41

(66 - 62.11)^2 = (3.89)^2 ≈15.13

(70 - 62.11)^2 = (7.89)^2 ≈62.25

(72 - 62.11)^2 = (9.89)^2 ≈97.81

(70 - 62.11)^2 = (7.89)^2 ≈62.25

(66 - 62.11)^2 = (3.89)^2 ≈15.13

(67 - 62.11)^2 = (4.89)^2 ≈23.91

Sum of squared differences: 23.91 + 534.07 + 404.41 + 15.13 + 62.25 + 97.81 + 62.25 + 15.13 + 23.91 = let's add:

23.91 + 534.07 = 557.98

557.98 + 404.41 = 962.39

962.39 + 15.13 = 977.52

977.52 + 62.25 = 1039.77

1039.77 + 97.81 = 1137.58

1137.58 + 62.25 = 1199.83

1199.83 + 15.13 = 1214.96

1214.96 + 23.91 = 1238.87

Population variance \(\sigma^{2}=\frac{1238.87}{9}\approx137.65\)

Population standard deviation \(\sigma=\sqrt{137.65}\approx11.73\)

Now find \( \mu\pm2\sigma\):

Lower bound: \(62.11 - 2\times11.73 = 62.11 - 23.46 = 38.65\)

Upper bound: \(62.11 + 2\times11.73 = 62.11 + 23.46 = 85.57\)

Now check each data point:

• 67: 67 is between 38.65 and 85.57: yes
• 39: 39 > 38.65: yes (39 - 38.65 = 0.35, so inside)
• 42: yes
• 66: yes
• 70: yes
• 72: 72 < 85.57: yes
• 70: yes
• 66: yes
• 67: yes

So all 9 data points? Wait 39: 39 is greater than 38.65, so yes. Let's check:

38.65 < 39 < 85.57: yes. 42: yes. 66: yes. 70: yes. 72: yes. 70: yes. 66: yes. 67: yes. So all 9 data points are within \( \mu\pm2\sigma\)? Wait that can't be. Wait 72 is 72, upper bound is 85.57, so yes. 39 is 39, lower bound 38.65, so 39 > 38.65, so yes. So all 9 data points are within 2 standard deviations?

Wait let's recalculate the mean correctly. Data points: 67, 39, 42, 66, 70, 72, 70, 66, 67. Let's sum again:

67 + 39 = 106

106 + 42 = 148

148 + 66 = 214

214 + 70 = 284

284 + 72 = 356

356 + 70 = 426

426 + 66 = 492

492 + 67 = 559. Yes, sum is 559. Mean is 559/9 ≈62.111...

Now calculate each deviation:

67 - 62.111 ≈4.889, squared ≈23.90

39 - 62.111 ≈-23.111, squared ≈534.12

42 - 62.111 ≈-20.111, squared ≈404.45

66 - 62.111 ≈3.889, squared ≈15.12

70 - 62.111 ≈7.889, squared ≈62.24

72 - 62.111 ≈9.889, squared ≈97.80

70 - 62.111 ≈7.889, squared ≈62.24

66 - 62.111 ≈3.889, squared ≈15.12

67 - 62.111 ≈4.889, squared ≈23.90

Sum of squared deviations: 23.90 + 534.12 = 558.02; +404.45 = 962.47; +15.12 = 977.59; +62.24 = 1039.83; +97.80 = 1137.63; +62.24 = 1199.87; +15.12 = 1214.99; +23.90 = 1238.89.

Population variance: 1238.89 / 9 ≈137.654

Population standard deviation: sqrt(137.654) ≈11.732

Now 2*sigma ≈23.464

Mean - 2*sigma ≈62.111 - 23.464 ≈38.647

Mean + 2*sigma ≈62.111 + 23.464 ≈85.575

Now check each data point:

• 67: 38.647 < 67 < 85.575: yes
• 39: 38.647 < 39 < 85.575: yes (39 - 38.647 ≈0.353)
• 42: yes
• 66: yes
• 70: yes
• 72: 72 < 85.575: yes
• 70: yes
• 66: yes
• 67: yes

So all 9 data points are within 2 standard deviations of the mean. Wait, that's because 39 is just above the lower bound (38.647), so it's included. So the number of data points within the range is 9. Therefore, the percentage is (9/9)*100% = 100%? But that seems off. Wait maybe I made a mistake in the data set. Let's list the data again: 67, 39, 42, 66, 70, 72, 70, 66, 67. Let's sort them: 39, 42, 66, 66, 67, 67, 70, 70, 72.

Now, mean is ~62.11, sigma ~11.73. So 2 sigma below mean is ~38.65, 2 sigma above is ~85.58. All data points are between 39 and 72, which is within 38.65 and 85.58. So yes, all 9 data points are within 2 standard deviations. Therefore, the percentage is 100%? Wait but that seems unusual. Wait maybe the initial sum was wrong. Let's add again: 67 + 39 = 106; +42 = 148; +66 = 214; +70 = 284; +72 = 356; +70 = 426; +66 = 492; +67 = 559. Yes, sum is 559. 559/9 = 62.111...

Wait, maybe the problem is a population, so we use population standard deviation. Let's check with a calculator approach. Alternatively, maybe the empirical rule, but the empirical rule is for normal distributions, but here we calculate directly.

Wait, let's use a calculator for the data set: 39, 42, 66, 66, 67, 67, 70, 70, 72.

Mean: (39 + 42 + 66 + 66 + 67 + 67 + 70 + 70 + 72)/9 = (39+42=81; 66+66=132; 67+67=134; 70+70=140; 72. So 81+132=213; +134=347; +140=487; +72=559. 559/9≈62.11.

Population standard deviation:

Using the formula \(\sigma=\sqrt{\frac{\sum (x_i - \mu)^2}{N}}\)

We calculated \(\sum (x_i - \mu)^2 = 1238.89\), so \(\sigma=\sqrt{1238.89/9}=\sqrt{137.654}\approx11.73\)

Now, 2*sigma≈23.46. So the interval is (62.11 - 23.46, 62.11 + 23.46) = (38.65, 85.57). Now, all data points: