The heights of 8 fourth graders are listed in inches: {50, 51, 55, 52, 59, 54, 54, 53}. Find the mean, median, mode, variance, and range. Also, do you think this sample might have come from a normal population? Why or why not?
Don't you have a calculator to do all this?
Just in case you don't understand the math terms there:
http://www.purplemath.com/modules/meanmode.htm
To find the mean, median, mode, variance, and range of the given data set, you can follow these steps:
1. Mean: The mean is the average of all the numbers in the data set. To find the mean, add up all the heights and then divide the sum by the total number of values. In this case, you have 8 heights, so the mean is (50 + 51 + 55 + 52 + 59 + 54 + 54 + 53) / 8 = 428 / 8 = 53.5 inches.
2. Median: The median is the middle value of the data set when arranged in ascending or descending order. To find the median, first arrange the heights in ascending order: {50, 51, 52, 53, 54, 54, 55, 59}. Since there are 8 values, the median will be the average of the 4th and 5th values, which are 53 and 54. Therefore, the median is (53 + 54) / 2 = 53.5 inches.
3. Mode: The mode is the value(s) that appear most frequently in the data set. In this case, the heights 54 inches appear twice, which is more often than any other height. Hence, the mode is 54 inches.
4. Variance: Variance measures the spread or dispersion of the data set. It is calculated by taking the average of the squared differences between each value and the mean. To find the variance, subtract the mean from each height, square the result, and find the average of these squared differences. The calculations are as follows:
(50 - 53.5)^2 + (51 - 53.5)^2 + (55 - 53.5)^2 + (52 - 53.5)^2 + (59 - 53.5)^2 + (54 - 53.5)^2 + (54 - 53.5)^2 + (53 - 53.5)^2 = 16.875
Next, divide the sum by the number of values minus 1 (since this is a sample, not the whole population) to get the variance: 16.875 / (8 - 1) = 3.375 inches squared.
5. Range: The range is simply the difference between the largest and smallest values in the data set. In this case, the largest value is 59 inches, and the smallest value is 50 inches. Therefore, the range is 59 - 50 = 9 inches.
As for whether the sample might have come from a normal population, we can examine the data set and qualitatively assess its distribution. In a normal distribution, data tends to cluster around the mean and follow a symmetric bell-shaped curve. From the given heights, it's difficult to say definitively whether they come from a normal population without further analysis or a larger sample size. However, visually observing the data set, we can notice slight skewness, caused by the presence of outliers (e.g., 50 and 59 inches). This suggests that the sample may not represent a normal population. To draw a more accurate conclusion, statistical tests or a larger sample size would be necessary.