Mean is 64.72 (approximately)
Median is 69.0 (approximately)
Mode is 64.0
Mean:
Calculate the class marks: We need the class marks for each class to calculate the mean. We can use the following formula:
class_mark = (cf - f/2) * (upper_class_limit - last_class_mark) / (cf.max() - f.max()) + last_class_mark
where:
cf is the cumulative frequency
f is the frequency
upper_class_limit is the upper limit of the class interval
last_class_mark is the mark of the last class
Using this formula, we can calculate the class marks for each class as shown below:
Class Frequency (f) Cumulative Frequency (cf) Class Mark
1 6 6 58.0
2 10 16 61.0
3 10 26 64.0
4 6 32 67.0
5 10 42 70.0
Calculate the mean: Now that we have the class marks, we can simply calculate the mean by multiplying each class mark by its frequency, summing the products, and then dividing by the total number of observations (50 in this case).
mean = sum(f * class_mark) / N
where:
f is the frequency of each class
class_mark is the class mark of each class
N is the total number of observations
Plugging in the values, we get:
mean = (6 * 58 + 10 * 61 + 10 * 64 + 6 * 67 + 10 * 70) / 50
mean ≈ 64.72
Therefore, the estimated mean is approximately 64.72.
Median:
Find the median position: The median is the observation that cuts the data set in half when arranged in ascending order. Since we have 50 observations, the median position is the 25th observation (N/2).
Identify the class containing the median: We need to find the class interval that contains the 25th observation. We can do this by looking at the cumulative frequencies. The class with a cumulative frequency greater than or equal to 25 will contain the median.
In this case, the 25th observation falls within the third class interval (cf = 26).
Calculate the median: Since we have grouped data, we cannot calculate the exact median. Instead, we can estimate it using the following formula:
median = L + (n/2 - cf_previous) * w
where:
L is the lower limit of the class interval containing the median
n is the total number of observations
cf_previous is the cumulative frequency of the class before the median class
w is the width of the class interval
Plugging in the values, we get:
median = 64 + (50/2 - 26) * 3
median ≈ 69.0
Therefore, the estimated median is approximately 69.0.
Mode:
The mode is the most frequent value. From the data, we can see that the third class interval has the highest frequency (10). Therefore, the mode is 64.0.