The heights in inches of 14 randomly selected adult males in LA are listed as:
71, 67, 70, 59, 71, 68, 67, 71, 80, 68, 74, 69, 72, 68
Display the data in a stem and leaf plot. find the mean, median, mode, range, variance, and standard deviation.
Mode is most frequently occurring score.
For median, arrange in order of value and choose middle-most value.
Range is highest-lowest values.
To find the mean, sum scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
I'll let you do the calculations.
To create a stem and leaf plot for the given data, follow these steps:
Step 1: Separate each data point into a "stem" and a "leaf" component.
Step 2: Write the stems in ascending order vertically down the left-hand side of the plot.
Step 3: Next to each stem, write the corresponding leaf(s) in ascending order.
For the given data:
59, 67, 67, 68, 68, 68, 69, 70, 71, 71, 71, 72, 74, 80
The stem and leaf plot is:
5 | 9
6 | 7 7 8 8 8 9
7 | 0 1 1 2 4
8 | 0
Now, let's calculate the various statistical measures:
Mean:
To find the mean, sum up all the data points and divide by the total number of data points.
Mean = (71 + 67 + 70 + 59 + 71 + 68 + 67 + 71 + 80 + 68 + 74 + 69 + 72 + 68) / 14
Mean = 948 / 14
Mean ≈ 67.71 inches
Median:
To find the median, we need to rearrange the data in ascending order and then find the middle value.
Arranging the data in ascending order: 59, 67, 67, 68, 68, 68, 69, 70, 71, 71, 71, 72, 74, 80
Median = (69 + 70) / 2
Median = 69.5 inches
Mode:
The mode is the value(s) that appear most frequently in the data set.
In this case, the mode is 68 inches because it appears three times, while all other values only appear once or twice.
Range:
The range is the difference between the largest and smallest values in the data set.
Range = Largest value - Smallest value
Range = 80 - 59
Range = 21 inches
Variance:
Variance measures how spread out the data is from the mean.
To calculate the variance, follow these steps:
1. Find the difference between each data point and the mean.
2. Square each difference.
3. Sum up all the squared differences.
4. Divide the sum by the number of data points.
Let's calculate it:
Step 1: Calculate the differences (data point - mean):
2.29, -0.71, 2.29, -8.71, 2.29, 0.29, -0.71, 2.29, 12.29, 0.29, 6.29, 1.29, 4.29, 0.29
Step 2: Square each difference:
5.2441, 0.5041, 5.2441, 75.6641, 5.2441, 0.0841, 0.5041, 5.2441, 151.2441, 0.0841, 39.5641, 1.6641, 18.3241, 0.0841
Step 3: Sum up all the squared differences:
Sum = 343.64
Step 4: Divide the sum by the number of data points:
Variance = Sum ÷ Number of data points
Variance = 343.64 ÷ 14
Variance ≈ 24.54 square inches
Standard Deviation:
The standard deviation is the square root of the variance. It represents the average amount of deviation from the mean.
Standard Deviation = √Variance
Standard Deviation ≈ √24.54
Standard Deviation ≈ 4.95 inches
So, the statistics for the given data are:
Mean ≈ 67.71 inches
Median = 69.5 inches
Mode = 68 inches
Range = 21 inches
Variance ≈ 24.54 square inches
Standard Deviation ≈ 4.95 inches