Question

question
for the following set of data, find the percentage of data within 1 population standard deviation of the mean, to the nearest 10th of a percent.
72, 69, 82, 74, 69, 78, 68
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Explanation:

Step1: Calculate the mean

First, find the sum of the data set: \(72 + 69 + 82 + 74 + 69 + 78 + 68 = 512\). There are 7 data points, so the mean \(\mu=\frac{512}{7}\approx73.14\).

Step2: Calculate the population standard deviation

For each data point \(x_i\), find \((x_i - \mu)^2\):
- \((72 - 73.14)^2\approx1.30\)
- \((69 - 73.14)^2\approx17.14\)
- \((82 - 73.14)^2\approx78.49\)
- \((74 - 73.14)^2\approx0.74\)
- \((69 - 73.14)^2\approx17.14\)
- \((78 - 73.14)^2\approx23.62\)
- \((68 - 73.14)^2\approx26.42\)

Sum these squared differences: \(1.30 + 17.14 + 78.49 + 0.74 + 17.14 + 23.62 + 26.42 = 164.85\). Then, the population variance \(\sigma^2=\frac{164.85}{7}\approx23.55\), and the population standard deviation \(\sigma=\sqrt{23.55}\approx4.85\).

Step3: Find the range within 1 standard deviation

The range is \(\mu - \sigma\) to \(\mu + \sigma\), so \(73.14 - 4.85 = 68.29\) to \(73.14 + 4.85 = 77.99\).

Step4: Count data points in the range

Check which data points are in [68.29, 77.99]: 72, 69, 74, 69, 68 (wait, 68 is 68 < 68.29? Wait, 68.29 is approximately 68.3, so 68 is below. Wait, recalculate: 68.29, so 69, 72, 74, 69, 78? No, 78 is 78 > 77.99. Wait, let's list the data: 72 (in), 69 (in), 82 (out), 74 (in), 69 (in), 78 (out), 68 (out? 68 vs 68.29: 68 < 68.29, so out). Wait, maybe I miscalculated the mean. Wait, sum: 72+69=141, +82=223, +74=297, +69=366, +78=444, +68=512. 512/7=73.142857... Correct. Standard deviation: let's use a better way. Maybe use calculator steps. Alternatively, use the empirical rule or count correctly. Wait, maybe my standard deviation calculation is off. Let's use the formula for population standard deviation: \(\sigma=\sqrt{\frac{\sum(x_i - \mu)^2}{N}}\). Let's recalculate \(\sum(x_i - \mu)^2\):

\(72 - 73.142857 = -1.142857\), squared: 1.306122

\(69 - 73.142857 = -4.142857\), squared: 17.163265

\(82 - 73.142857 = 8.857143\), squared: 78.450000

\(74 - 73.142857 = 0.857143\), squared: 0.734694

\(69 - 73.142857 = -4.142857\), squared: 17.163265

\(78 - 73.142857 = 4.857143\), squared: 23.591837

\(68 - 73.142857 = -5.142857\), squared: 26.448980

Sum these: 1.306122 + 17.163265 = 18.469387; +78.45 = 96.919387; +0.734694 = 97.654081; +17.163265 = 114.817346; +23.591837 = 138.409183; +26.448980 = 164.858163. Then \(\sigma^2 = 164.858163 / 7 ≈23.551166\), \(\sigma≈4.853\). So \(\mu - \sigma≈73.1429 - 4.853≈68.2899\), \(\mu + \sigma≈73.1429 + 4.853≈77.9959\). Now check each data point:

- 72: between 68.29 and 77.9959: yes.
- 69: yes.
- 82: 82 > 77.9959: no.
- 74: yes.
- 69: yes.
- 78: 78 > 77.9959? 78 vs 77.9959: 78 is slightly above, so no.
- 68: 68 < 68.2899? 68 vs 68.29: 68 is below, so no. Wait, that's 4 data points (72, 69, 74, 69) out of 7? Wait, 72, 69, 74, 69: that's 4. Wait, maybe I made a mistake in the range. Wait, population standard deviation, so the formula is correct. Wait, maybe the data is 7 values: 72,69,82,74,69,78,68. Let's list their distances from mean:

72: 73.14 - 72 = 1.14 (below mean)
69: 73.14 - 69 = 4.14 (below)
82: 82 - 73.14 = 8.86 (above)
74: 74 - 73.14 = 0.86 (above)
69: 4.14 (below)
78: 78 - 73.14 = 4.86 (above)
68: 73.14 - 68 = 5.14 (below)

So within 1 standard deviation (≈4.85) from mean: above mean: 0.86 (74) is within 4.85, 4.86 (78) is above 4.85 (since 4.86 > 4.85), so 78 is out. Below mean: 1.14 (72) is within 4.85, 4.14 (69) is within 4.85, 5.14 (68) is above 4.85 (so out). So below mean: 72 (1.14), 69 (4.14), 69 (4.14) – wait, 72 is above mean? Wait, mean is 73.14, so 72 is below mean (73.14 - 72 = 1.14). So below mean: 72 (1.14), 69 (4.14), 69 (4.14), 68 (5.14). Above mean: 74 (0.86), 78 (4.86), 82 (8.86). So within 1 SD (4.85) from mean:

Below mean: distances ≤4.85: 1.14 (72), 4.14 (69), 4.14 (69) – that's 3.

Above mean: distances ≤4.85: 0.86 (74), 4.86 (78) – wait, 4.86 is slightly more than 4.85, so 78 is out. So 0.86 (74) is in. So total in: 72, 69, 69, 74 – that's 4. Wait, but 4 out of 7 is approximately 57.1%, but that seems low. Wait, maybe I miscalculated the standard deviation. Let's use a calculator for population standard deviation.

Using a calculator:

Data: 72, 69, 82, 74, 69, 78, 68

Sum: 72+69=141, +82=223, +74=297, +69=366, +78=444, +68=512. Mean: 512/7≈73.142857.

Population variance: sum of (x - mean)^2 / 7.

Calculate each (x - mean)^2:

(72 - 73.142857)^2 = (-1.142857)^2 ≈1.306122

(69 - 73.142857)^2 = (-4.142857)^2 ≈17.163265

(82 - 73.142857)^2 = (8.857143)^2 ≈78.450000

(74 - 73.142857)^2 = (0.857143)^2 ≈0.734694

(69 - 73.142857)^2 = (-4.142857)^2 ≈17.163265

(78 - 73.142857)^2 = (4.857143)^2 ≈23.591837

(68 - 73.142857)^2 = (-5.142857)^2 ≈26.448980

Sum these: 1.306122 + 17.163265 = 18.469387; +78.45 = 96.919387; +0.734694 = 97.654081; +17.163265 = 114.817346; +23.591837 = 138.409183; +26.448980 = 164.858163.

Variance: 164.858163 / 7 ≈23.551166.

Standard deviation: √23.551166 ≈4.853.

Now, the range is mean - SD to mean + SD: 73.142857 - 4.853 ≈68.2899, 73.142857 + 4.853 ≈77.9959.

Now check each data point:

- 72: 68.2899 ≤72 ≤77.9959: yes (72 is between 68.29 and 78.00)
- 69: yes (69 between 68.29 and 78.00)
- 82: 82 >77.9959: no
- 74: yes (74 between 68.29 and 78.00)
- 69: yes
- 78: 78 >77.9959: no (78 is 78.0, which is just above 77.9959, so no)
- 68: 68 <68.2899: no (68 is 68.0, below 68.29)

So the data points in the range are: 72, 69, 74, 69 – that's 4 data points. Wait, but 4 out of 7 is 4/7≈57.14%, which is approximately 57.1%. But wait, maybe I made a mistake with 78: 77.9959 is almost 78, so 78 is very close. Let's check 78 - 73.142857 = 4.857143, which is slightly more than SD (4.853), so 78 is outside. 68: 73.142857 - 68 = 5.142857, which is more than SD (4.853), so 68 is outside. 82 is way outside. So 4 data points: 72, 69, 74, 69. Wait, but 72, 69, 74, 69 – that's 4. Wait, but 72, 69, 74, 69, and is 78 included? 78 is 78, which is 78 - 73.14 = 4.86, which is 4.86 > 4.85, so no. 68: 5.14 > 4.85, so no. 82: 8.86 > 4.85, so no. So 4 out of 7. 4/7≈0.5714, so 57.1%. But wait, maybe the problem is using sample standard deviation? No, the question says "population standard deviation", so we used population.

Wait, maybe I made a mistake in the number of data points. Let's count: 72,69,82,74,69,78,68 – that's 7 data points. Correct.

Wait, another approach: maybe the mean is calculated as (72+69+82+74+69+78+68)/7 = 512/7 ≈73.14, correct.

Alternatively, maybe the question is using the empirical rule, but the empirical rule is for normal distributions, and this data may not be normal, but we have to calculate.

Wait, maybe I missed a data point. Let's list again: 72,69,82,74,69,78,68 – 7 numbers. Yes.

So 4 out of 7 is approximately 57.1%, which is 57.1 when rounded to the nearest tenth of a percent.

Answer:

57.1%