given the group of numbers: 8,9,10,10,10,11,11,11,12,13. which of the following are the same?
a)range and mean,b)mode and median,
c)standard deviation and mode
d)mean and median, e)none
Have you calculated any of these to make a comparison? For example, range = 5, so you can eliminate A. Do the values form a normal distribution? If so, what do you know about a normal distribution?
To determine which of the following pairs are the same, we can calculate each value and compare them:
a) Range and mean:
Range = Maximum value - Minimum value = 13 - 8 = 5
Mean = Sum of all values / Total number of values = (8+9+10+10+10+11+11+11+12+13) / 10 = 106 / 10 = 10.6
The range and mean are not the same.
b) Mode and median:
Mode is the value that appears most frequently, and median is the middle value when the numbers are arranged in ascending order.
Mode = 10 and 11 (both appear 3 times)
Median = (10 + 10) / 2 = 10
The mode and median are not the same.
c) Standard deviation and mode:
The standard deviation measures the dispersion of the values from the mean, while the mode is the value that appears most frequently.
To calculate the standard deviation:
1. Find the mean: 10.6
2. Subtract the mean from each value: 8-10.6, 9-10.6, 10-10.6, 10-10.6, 10-10.6, 11-10.6, 11-10.6, 11-10.6, 12-10.6, 13-10.6.
3. Square each result: (-2.6)^2, (-1.6)^2, (-0.6)^2, (-0.6)^2, (-0.6)^2, (0.4)^2, (0.4)^2, (0.4)^2, (1.4)^2, (2.4)^2.
4. Find the mean of the squared values: (6.76+2.56+0.36+0.36+0.36+0.16+0.16+0.16+1.96+5.76) / 10 = 1.48.
5. Take the square root of the mean: √1.48 = 1.22.
The standard deviation is approximately 1.22.
The standard deviation and mode are not the same.
d) Mean and median:
As calculated earlier:
Mean = 10.6
Median = 10
The mean and median are not the same.
e) None of the pairs are the same.
To determine which of the following pairs are the same in the given group of numbers (8, 9, 10, 10, 10, 11, 11, 11, 12, 13), we can calculate the different statistical measures. Let's go through each pair:
a) Range and mean:
- Range refers to the difference between the highest and lowest values in the dataset. In this case, the range is 13 (highest value) - 8 (lowest value) = 5.
- Mean is the average of all the numbers in the dataset. In this case, the mean is (8 + 9 + 10 + 10 + 10 + 11 + 11 + 11 + 12 + 13) / 10 = 10.5.
Since the range (5) and mean (10.5) are different, a) Range and mean are not the same.
b) Mode and median:
- Mode is the value(s) that appear most frequently in the dataset. In this case, the mode is 10 and 11, both occurring three times.
- Median is the middle value in the dataset when the numbers are arranged in ascending (or descending) order. In this case, after arranging the numbers in ascending order, the median is 10.5, which is the average of the two middle values (10 and 11).
Since the mode (10 and 11) and median (10.5) are different, b) Mode and median are not the same.
c) Standard deviation and mode:
- Standard deviation measures the amount of variation or dispersion in a dataset. Calculating the standard deviation requires several steps and calculations that are not straightforward to explain in a short answer. However, to determine whether the standard deviation is the same as the mode, we can compare their values.
- As we found previously, the mode in this dataset is 10 and 11.
Since the mode (10 and 11) and standard deviation cannot be determined without calculations, c) Standard deviation and mode cannot be determined.
d) Mean and median:
- As we found previously, the mean in this dataset is 10.5.
- Also, as we found earlier, the median in this dataset is 10.5.
Since the mean (10.5) and median (10.5) are the same, d) Mean and median are the same.
Therefore, the correct answer is d) Mean and median.