Let's go step-by-step to calculate the mean, median, and mode for the original set and for the two altered situations:

Original Set:

The data provided is: $20, 25, 25, 36, 42, 56, 69, 74, 98, 100$

1. Mean:

The mean is calculated by summing all the values and dividing by the number of values. $\text{Mean} = \frac{20 + 25 + 25 + 36 + 42 + 56 + 69 + 74 + 98 + 100}{10}$ $\text{Mean} = \frac{545}{10} = 54.5$

2. Median:

The median is the middle value when the data is sorted in ascending order. Since there are 10 values (an even number), the median is the average of the 5th and 6th values. The sorted data is: $20, 25, 25, 36, 42, 56, 69, 74, 98, 100$ The 5th value is 42, and the 6th value is 56. $\text{Median} = \frac{42 + 56}{2} = \frac{98}{2} = 49$

3. Mode:

The mode is the value that appears most frequently. In the data set: $25 \text{ appears twice},$ so the mode is 25.

Case 1: Score 98 Mistakenly Recorded as 78

Now the data becomes: $20, 25, 25, 36, 42, 56, 69, 74, 78, 100$

1. Mean:

$\text{Mean} = \frac{20 + 25 + 25 + 36 + 42 + 56 + 69 + 74 + 78 + 100}{10}$ $\text{Mean} = \frac{525}{10} = 52.5$

2. Median:

The sorted data is now: $20, 25, 25, 36, 42, 56, 69, 74, 78, 100$ The 5th and 6th values are still 42 and 56, so: $\text{Median} = \frac{42 + 56}{2} = 49$

3. Mode:

The mode remains the same, as 25 still appears twice, so: $\text{Mode} = 25$

Case 2: Score 98 Removed

Now the data becomes: $20, 25, 25, 36, 42, 56, 69, 74, 100$

1. Mean:

$\text{Mean} = \frac{20 + 25 + 25 + 36 + 42 + 56 + 69 + 74 + 100}{9}$ $\text{Mean} = \frac{447}{9} \approx 49.67$

2. Median:

The sorted data is: $20, 25, 25, 36, 42, 56, 69, 74, 100$ Since there are 9 values (an odd number), the median is the middle value, which is the 5th value: $\text{Median} = 42$

3. Mode:

The mode remains 25, as it appears twice, so: $\text{Mode} = 25$

Summary of Changes:

1. Original Set:
   • Mean = 54.5, Median = 49, Mode = 25
2. With Error (98 recorded as 78):
   • Mean = 52.5, Median = 49, Mode = 25
3. With 98 Removed:
   • Mean = 49.67, Median = 42, Mode = 25

Would you like more detailed steps or have any questions?

Related Questions:

1. How does an outlier (like 98) affect the mean compared to the median?
2. If another score was incorrectly recorded, how would that impact the mode?
3. How would you calculate the range of the dataset in each scenario?
4. Can you calculate the variance and standard deviation for the original dataset?
5. What would happen if there were multiple modes in the dataset?

Tip:

When analyzing a data set, always verify for outliers or errors, as they can significantly affect statistical measures like the mean.