The modal age is 9 years old, the median age is 9.5 years old, the mean age is 8.8 years old, and the standard deviation is approximately 1.49 years old.
Step-by-step explanation:
To find the modal age, we need to find the age group with the highest frequency. In this case, the age group with the highest frequency is 9 years old, with a frequency of 11. Therefore, the modal age is 9 years old.
To find the median age, we need to arrange the ages in ascending order. The ages are already arranged in ascending order in the table. Since there are 40 pupils in total, the median age will be the average of the 20th and 21st ages, which are 9 and 10 years old, respectively. Therefore, the median age is (9 + 10)/2 = 9.5 years old.
To find the mean age, we need to calculate the sum of the products of each age and its corresponding frequency. The calculation is as follows: (6*4 + 7*6 + 8*10 + 9*11 + 10*8 + 11*1) / 40 = 8.8 years old. Therefore, the mean age is 8.8 years old.
To find the standard deviation, first find the variance by calculating the sum of the squared differences between each age and the mean, multiplied by its frequency, and then divide by the total frequency. The calculation is as follows: ((6-8.8)^2 * 4 + (7-8.8)^2 * 6 + (8-8.8)^2 * 10 + (9-8.8)^2 * 11 + (10-8.8)^2 * 8 + (11-8.8)^2 * 1) / 40 = 2.23. Finally, take the square root of the variance to find the standard deviation. Therefore, the standard deviation is approximately 1.49 years old.