The heights in inches of 18 randomly selected adult males in LA are listed as: 70, 69, 72, 57, 70, 66, 69, 73, 80, 68, 71, 68, 72, 67, 58, 74, 81, 72.
Display the data in a stem-and-leaf plot.
1. Find the mean.
2. Find the median.
3. Find the mode.
4. Find the range.
5. Find the variance.
6. Find the standard deviation.
Stem and leaf plots are done in order. The stems are the tens places and the leaves are the ones.
Stem: Leaf:
5 7 8
For 57, 58. Continue doing this with the remaining numbers in order.
The mean is the average. Add all of your numbers up and divide by the amount of numbers you have. For the median, put the numbers in order from greatest to least (or vice versa) and find the number in the middle. If there are two numbers in the middle, take the average of them. The mode is the number used the most. The range can be found by subtracting the smallest number from the highest number. Sorry, I forget variance and standard deviation. If you want to post your answers, I'll be happy to check them.
To create a stem-and-leaf plot for the given data, follow these steps:
Step 1: Separate the data into stems and leaves.
- The stem represents the tens digit of each number.
- The leaf represents the ones digit of each number.
Here is the stem-and-leaf plot for the given data:
5 | 7 8 8
6 | 6 7 8 9
7 | 0 0 1 2 2 3 4
8 | 0 1
Now, let's calculate the statistical measures for the data:
1. Mean:
The mean is calculated by summing up all the values and dividing it by the total number of values.
Mean = (70 + 69 + 72 + 57 + 70 + 66 + 69 + 73 + 80 + 68 + 71 + 68 + 72 + 67 + 58 + 74 + 81 + 72) / 18
Mean = 1241 / 18
Mean ≈ 68.94
2. Median:
To find the median, the data must be arranged in numerical order, and then the middle value is selected. If there is an even number of values, the median is the average of the two middle values.
Arranged data: 57, 58, 66, 67, 68, 68, 69, 69, 70, 70, 71, 72, 72, 73, 74, 80, 81
Since there are 18 values, the median is the average of the 9th and 10th values.
Median = (69 + 70) / 2
Median = 139 / 2
Median = 69.5
3. Mode:
The mode represents the value(s) that appear(s) most frequently in the dataset.
In this dataset, there is no mode. Each value appears only once.
4. Range:
The range is the difference between the largest and smallest values in the dataset.
Range = Largest value - Smallest value
Range = 81 - 57
Range = 24
5. Variance:
Variance measures how spread out the data is from the mean.
For sample variance, the formula is:
Variance = Σ( (xi - mean)^2 ) / (n - 1)
Using the given data and the previously calculated mean, we can calculate the variance.
Variance = ( (70-68.94)^2 + (69-68.94)^2 + ... + (81-68.94)^2 ) / (18-1)
Variation = Σ( (xi - mean)^2 ) / (n - 1)
Variance ≈ 41.99
6. Standard Deviation:
The standard deviation is the square root of the variance.
Standard Deviation = √Variance
Standard Deviation ≈ √41.99
Standard Deviation ≈ 6.48