Question

the table below shows the scores on a math test.
david 75 aaron 75
ashley 60 gina 75
andrew 75 kei 95
monica 100 anna 60
lucas 75 jennifer 65
find the standard deviation. round to the nearest tenth when necessary.
a. 157.25 \tc. 9.1
b. 12.5 \td. 12.3
please select the best answer from the choices provided

Explanation:

Step1: List all scores

Scores: 75, 60, 75, 100, 75, 75, 75, 95, 60, 65 (total 10 scores)

Step2: Calculate the mean ($\mu$)

Sum of scores: \(75 + 60 + 75 + 100 + 75 + 75 + 75 + 95 + 60 + 65 = 750\)
Mean: \(\mu=\frac{750}{10} = 75\)

Step3: Calculate squared differences from mean

For each score \(x\), compute \((x - \mu)^2\):
- \(75 - 75 = 0\), \((0)^2 = 0\) (5 times, since 75 appears 5 times)
- \(60 - 75 = -15\), \((-15)^2 = 225\) (2 times)
- \(100 - 75 = 25\), \((25)^2 = 625\) (1 time)
- \(95 - 75 = 20\), \((20)^2 = 400\) (1 time)
- \(65 - 75 = -10\), \((-10)^2 = 100\) (1 time)

Step4: Calculate sum of squared differences

Sum: \(5\times0 + 2\times225 + 625 + 400 + 100 = 0 + 450 + 625 + 400 + 100 = 1575\)

Step5: Calculate variance ($\sigma^2$)

Variance: \(\sigma^2=\frac{1575}{10}=157.5\) (Wait, earlier miscalculation in sum? Wait, let's recalculate sum of scores: 75 (David) + 60 (Ashley) +75 (Andrew)+100 (Monica)+75 (Lucas)+75 (Aaron)+75 (Gina)+95 (Kei)+60 (Anna)+65 (Jennifer). Let's add again: 75+60=135; 135+75=210; 210+100=310; 310+75=385; 385+75=460; 460+75=535; 535+95=630; 630+60=690; 690+65=755? Wait, I made a mistake earlier. Let's recalculate sum:

David:75, Ashley:60 (75+60=135), Andrew:75 (135+75=210), Monica:100 (210+100=310), Lucas:75 (310+75=385), Aaron:75 (385+75=460), Gina:75 (460+75=535), Kei:95 (535+95=630), Anna:60 (630+60=690), Jennifer:65 (690+65=755). Oh! I had a mistake in sum. So mean is \(755/10 = 75.5\)

Now recalculate squared differences with mean 75.5:

- For 75 (5 times): \(75 - 75.5 = -0.5\), \((-0.5)^2 = 0.25\). Sum for these: \(5\times0.25 = 1.25\)
- For 60 (2 times): \(60 - 75.5 = -15.5\), \((-15.5)^2 = 240.25\). Sum: \(2\times240.25 = 480.5\)
- For 100: \(100 - 75.5 = 24.5\), \((24.5)^2 = 600.25\). Sum: 600.25
- For 95: \(95 - 75.5 = 19.5\), \((19.5)^2 = 380.25\). Sum: 380.25
- For 65: \(65 - 75.5 = -10.5\), \((-10.5)^2 = 110.25\). Sum: 110.25

Now total sum of squared differences: \(1.25 + 480.5 + 600.25 + 380.25 + 110.25 = 1572.5\)

Variance: \(\sigma^2=\frac{1572.5}{10}=157.25\)

Standard deviation: \(\sigma=\sqrt{157.25}\approx12.5\) (Wait, but wait, let's check again. Wait, maybe my initial count of scores was wrong. Let's list all scores:

David:75

Ashley:60

Andrew:75

Monica:100

Lucas:75

Aaron:75

Gina:75

Kei:95

Anna:60

Jennifer:65

That's 10 scores. Let's count the frequency of each score:

75: David, Andrew, Lucas, Aaron, Gina → 5 times

60: Ashley, Anna → 2 times

100: Monica →1

95: Kei →1

65: Jennifer →1

Total: 5+2+1+1+1=10. Correct.

Now sum of scores: 5×75 + 2×60 + 100 + 95 + 65 = 375 + 120 + 100 + 95 + 65 = 375+120=495; 495+100=595; 595+95=690; 690+65=755. Correct. Mean is 755/10=75.5.

Now squared differences:

For 75 (x=75, μ=75.5): (75-75.5)²=(-0.5)²=0.25. 5 times: 5×0.25=1.25

For 60 (x=60, μ=75.5): (60-75.5)²=(-15.5)²=240.25. 2 times: 2×240.25=480.5

For 100 (x=100, μ=75.5): (100-75.5)²=24.5²=600.25. 1 time: 600.25

For 95 (x=95, μ=75.5): (95-75.5)²=19.5²=380.25. 1 time: 380.25

For 65 (x=65, μ=75.5): (65-75.5)²=(-10.5)²=110.25. 1 time: 110.25

Sum of squared differences: 1.25 + 480.5 = 481.75; 481.75 + 600.25 = 1082; 1082 + 380.25 = 1462.25; 1462.25 + 110.25 = 1572.5. Correct.

Variance: 1572.5 / 10 = 157.25

Standard deviation: √157.25 ≈ 12.5 (since 12.5²=156.25, 12.6²=158.76, so 157.25 is between them, closer to 12.5? Wait, 12.5²=156.25, 12.5^2=156.25, 12.5^2 + 1 = 157.25, so √157.25 = √(156.25 + 1) ≈ 12.5 + 1/(2×12.5) = 12.5 + 0.04 = 12.54, which rounds to 12.5.

Wait, but the options include 12.5 (option b) and 12.3 (option d). Wait, maybe I made a mistake in mean? Wait, let's check sum again:

David:75

Ashley:60 (75+60=135)

Andrew:75 (135+75=210)

Monica:100 (210+100=310)

Lucas:75 (310+75=385)

Aaron:75 (385+75=460)

Gina:75 (460+75=535)

Kei:95 (535+95=630)

Anna:60 (630+60=690)

Jennifer:65 (690+65=755). Yes, 755. Mean 75.5.

Wait, maybe the problem is using population standard deviation (divide by N) or sample (divide by N-1)? Wait, the problem says "standard deviation" without specifying, but in math tests, sometimes it's population. Wait, but let's check the options. Option b is 12.5, which matches our calculation.

Wait, maybe I made a mistake in the number of scores. Let's count again: David, Ashley, Andrew, Monica, Lucas, Aaron, Gina, Kei, Anna, Jennifer. 10 students. Correct.

So standard deviation is √157.25 ≈12.5.

Answer:

b. 12.5