The table below gives the number of hours spent studying by a group of students over a week. Calculate the mode and standard deviation of the study hours.
Study Hours
Frequency (f)
1 - 5
5
6 - 10
8
11 - 15
4
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We have a frequency distribution table showing study hours for a group of students. Five students studied 1 to 5 hours, eight students studied 6 to 10 hours, and four students studied 11 to 15 hours. We need to calculate the mode and standard deviation of this data.
To find the mode, we first identify the modal class, which is the class with the highest frequency. Here, the 6 to 10 hours class has frequency 8, making it the modal class. We then apply the mode formula for grouped data. The lower boundary L is 5.5, f1 is 8, f0 is 5, f2 is 4, and class width w is 5. Substituting these values, we get mode equals 7.64 hours.
To calculate the standard deviation, we first find the midpoint of each class interval. For class 1 to 5, the midpoint x is 3. For class 6 to 10, x is 8. For class 11 to 15, x is 13. Next, we calculate f times x and f times x squared for each class. This gives us the sums we need for the standard deviation formula.
Now we calculate the standard deviation. First, we find the mean by dividing the sum of f times x by N, which gives us 131 divided by 17, equals 7.71 hours. Next, we calculate the variance using the formula: sigma squared equals the sum of f times x squared divided by N, minus the mean squared. This gives us 72.53 minus 59.38, equals 13.15. Finally, the standard deviation is the square root of the variance, which equals 3.63 hours.
To summarize our analysis: The mode of the study hours data is 7.64 hours, found using the mode formula for grouped data. The standard deviation is 3.63 hours, calculated using the variance formula. The modal class was 6 to 10 hours with the highest frequency of 8 students. The standard deviation tells us how the data is spread around the mean value.