Question

suppose a class obtained the following test scores.
score frequency
90 2
80 4
70 10
60 6
50 3
40 1
30 2
0 4
what is the median?

Explanation:

Step1: Calculate total number of data points

First, we sum up all the frequencies. The frequencies are 2, 4, 10, 6, 3, 1, 2, 4. So, \(2 + 4+10 + 6+3 + 1+2 + 4=\sum_{i}f_i = 32\).

Step2: Determine the position of the median

For a dataset with \(n = 32\) (even number of observations), the median is the average of the \(\frac{n}{2}\)-th and \((\frac{n}{2}+ 1)\)-th values. So, \(\frac{32}{2}=16\) and \(\frac{32}{2}+ 1 = 17\). We need to find the 16th and 17th values when the data is ordered.

Step3: Cumulative frequency calculation

We calculate cumulative frequencies:
- For score 90: cumulative frequency \(= 2\)
- For score 80: cumulative frequency \(= 2 + 4=6\)
- For score 70: cumulative frequency \(= 6+10 = 16\)
- For score 60: cumulative frequency \(= 16 + 6=22\) (and so on, but we can stop here as we found the 16th value)

The 16th value is 70 (since cumulative frequency reaches 16 at score 70). The 17th value is the next one, which is also 70 (because the next score is 60 but cumulative frequency at 70 is 16, so the 17th value is still in the 70 group? Wait, no. Wait, when cumulative frequency at 70 is 16, that means the first 16 values are up to 70. Then the 17th value is the first value of the next group? Wait, no, let's re - check. Wait, the data is ordered from highest to lowest? Wait, no, actually, scores are from 90 down to 0, but when we order the data (from lowest to highest, it's 0,30,40,50,60,70,80,90). Wait, maybe I ordered it wrong. Let's order the scores from lowest to highest: 0,30,40,50,60,70,80,90.

Now cumulative frequency from lowest to highest:
- Score 0: cumulative frequency \(= 4\)
- Score 30: cumulative frequency \(= 4 + 2=6\)
- Score 40: cumulative frequency \(= 6+1 = 7\)
- Score 50: cumulative frequency \(= 7+3 = 10\)
- Score 60: cumulative frequency \(= 10+6 = 16\)
- Score 70: cumulative frequency \(= 16+10 = 26\)

Ah, I had the order reversed before. So when ordering from lowest to highest:

- Score 0: frequency 4 (cumulative 4)
- Score 30: frequency 2 (cumulative 6)
- Score 40: frequency 1 (cumulative 7)
- Score 50: frequency 3 (cumulative 10)
- Score 60: frequency 6 (cumulative 16)
- Score 70: frequency 10 (cumulative 26)
- Score 80: frequency 4 (cumulative 30)
- Score 90: frequency 2 (cumulative 32)

Now, the 16th value: cumulative frequency at 60 is 16, so the 16th value is 60? Wait, no, cumulative frequency is the number of values less than or equal to the score. Wait, when we have score 0: 4 values (1 - 4), score 30: 5 - 6, score 40:7, score 50:8 - 10, score 60:11 - 16, score 70:17 - 26, etc. So the 16th value is 60, and the 17th value is 70. Then the median is the average of 60 and 70, which is \(\frac{60 + 70}{2}=65\)? Wait, no, wait, let's list the positions:

- Values 1 - 4: 0
- Values 5 - 6: 30
- Value 7: 40
- Values 8 - 10: 50
- Values 11 - 16: 60
- Values 17 - 26: 70
- Values 27 - 30: 80
- Values 31 - 32: 90

So the 16th value is 60 (last value of the 60 group), the 17th value is 70 (first value of the 70 group). Then median \(=\frac{60 + 70}{2}=65\). Wait, I made a mistake earlier in the order of scores. So the correct way is to order from lowest to highest.

Let's recalculate cumulative frequency from lowest to highest:

1. Score 0: frequency = 4, cumulative frequency = 4
2. Score 30: frequency = 2, cumulative frequency = 4+2 = 6
3. Score 40: frequency = 1, cumulative frequency = 6 + 1=7
4. Score 50: frequency = 3, cumulative frequency = 7+3 = 10
5. Score 60: frequency = 6, cumulative frequency = 10+6 = 16
6. Score 70: frequency = 10, cumulative frequency = 16+10 = 26
7. Score 80: frequency = 4, cumulative frequency = 26+4 = 30
8. Score 90: frequency = 2, cumulative frequency = 30+2 = 32

Now, the 16th term is the last term in the score 60 group (since cumulative frequency at 60 is 16), and the 17th term is the first term in the score 70 group. So the 16th value is 60, the 17th value is 70. Then median \(=\frac{60 + 70}{2}=65\).

Wait, but let's check again. The formula for median when \(n\) is even is \(\frac{X_{\frac{n}{2}}+X_{\frac{n}{2}+1}}{2}\). Here, \(n = 32\), so \(\frac{n}{2}=16\), \(\frac{n}{2}+1 = 17\). \(X_{16}\) is the 16th observation, \(X_{17}\) is the 17th. From the cumulative frequency, when we go from lowest to highest:

- Up to score 0: 4 observations (1 - 4)
- Up to score 30: 6 observations (1 - 6)
- Up to score 40: 7 observations (1 - 7)
- Up to score 50: 10 observations (1 - 10)
- Up to score 60: 16 observations (1 - 16)
- Up to score 70: 26 observations (1 - 26)

So the 16th observation is 60 (since it's the last one in the 60 group), the 17th observation is 70 (first one in the 70 group). Then median \(=\frac{60 + 70}{2}=65\).

Answer:

65