A sample of 5 observations has mean 32 and median 33. Later it is found that an observation was recorded incorrectly as 40 instead of 35 If we correct the data then which one of the following is correct.

Options

A. The mean and mediun remain the same
B. The median remains the same but the mean will decrease
C. The mean and median both will decrease
D. The mean remains the same hut median will decrease

Answer

To solve the problem step by step, we will analyze the given information and compute the necessary values after correcting the observation.

### Step 1: Understand the given data
We have a sample of 5 observations with:
- Mean = 32
- Median = 33

### Step 2: Calculate the total sum of the observations
The mean is calculated as the total sum of observations divided by the number of observations. Therefore, we can find the total sum (S) of the observations using the formula:

\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

Substituting the known values:

\[ 32 = \frac{S}{5} \]

Multiplying both sides by 5 gives:

\[ S = 32 \times 5 = 160 \]

### Step 3: List the observations
Since the median is 33, when the 5 observations are arranged in ascending order, the middle value (3rd observation) must be 33. Let’s denote the observations as \(a_1, a_2, a_3, a_4, a_5\) where \(a_3 = 33\).

### Step 4: Identify the incorrect observation
One of the observations was recorded incorrectly as 40 instead of 35. This means that the correct observation should replace 40 with 35.

### Step 5: Calculate the new sum after correction
The incorrect sum included 40, so we need to adjust the sum:

\[ \text{New Sum} = S - 40 + 35 = 160 - 40 + 35 = 155 \]

### Step 6: Calculate the new mean
Now, we can calculate the new mean with the corrected sum:

\[ \text{New Mean} = \frac{\text{New Sum}}{\text{Number of observations}} = \frac{155}{5} = 31 \]

### Step 7: Analyze the median
The median is determined by the order of the observations. Since the median was 33 and the correction involved changing 40 to 35, we need to check if the median changes. The new set of observations will still have 33 as the middle value because the order of the numbers will not change significantly enough to affect the median.

### Conclusion
- The new mean is 31 (which has decreased from 32).
- The median remains 33.

### Final Answer
The correct option is that the median remains the same, but the mean decreases.