Calculate the mode and mean for the following grouped data: Class Intervals and Frequencies (25-35: 21, 35-45: 23, 45-55: 14)

Here's how to find the mode and mean for the given data: Part 1: Calculate the Mode Step 1: Identify the modal class. The modal class is the class interval with the highest frequency. From the table, the highest frequency is 23, which corresponds to the class interval 35-45. So, the modal class is 35-45. Step 2: Extract values for the mode formula. The formula for the mode of grouped data is: Mode = L + ( (f_1 - f_0)/(2f_1 - f_0 - f_2) ) × h Where: L = 35 (lower limit of the modal class) h = 45 - 35 = 10 (size of the class interval) f_1 = 23 (frequency of the modal class) f_0 = 21 (frequency of the class preceding the modal class, 25-35) f_2 = 14 (frequency of the class succeeding the modal class, 45-55) Step 3: Substitute the values into the mode formula and calculate. Mode = 35 + ( (23 - 21)/(2(23) - 21 - 14) ) × 10 Mode = 35 + ( (2)/(46 - 35) ) × 10 Mode = 35 + ( (2)/(11) ) × 10 Mode = 35 + (20)/(11) Mode = 35 + 1.8181... Mode ≈ 36.82 Part 2: Calculate the Mean Step 1: Create a table to find the mid-points (x_i) and the product of frequency and mid-point (f_i x_i). | Class Interval | Frequency (f_i) | Mid-point (x_i) | f_i x_i | |:---------------|:------------------|:------------------|:----------| | 5-15 | 6 | 10 | 60 | | 15-25 | 11 | 20 | 220 | | 25-35 | 21 | 30 | 630 | | 35-45 | 23 | 40 | 920 | | 45-55 | 14 | 50 | 700 | | 55-65 | 5 | 60 | 300 | | Total | f_i = 80 | | f_i x_i = 2830 | Step 2: Apply the formula for the mean. The formula for the mean (direct method) is: x = ( f_i x_i)/( f_i) Step 3: Substitute the sums and calculate the mean. x = (2830)/(80) x = (283)/(8) x = 35.375 That's 2 down. 3 left today — send the next one.