41 (41 – 28.375) = 12.625 159.391
2 20 (20 – 28.375) = -8.375 70.141
3 35 (35 – 28.375) = 6.625 43.891
4 25 (25 – 28.375) = -3.375 11.391
5 23 (23 – 28.375) = -5.375 28.891
6 30 (30 – 28.375) = 1.625 2.641
7 21 (21 – 28.375) = -7.375 54.391
8 32 (32 – 28.375) = 3.625 13.141
Σ = 227 Σ = 0.000 Σ = 383.878
Let’s calculate the mean as our measure of central tendency by adding the individual ages of each officer and dividing by the number of officers. The calculation is 227/8 = 28.375 year
It is important to note from this calculation that the sum of the deviations of each score from the mean is equal to zero. When doing hand calculations of the mean deviation, variance, and standard deviation this is an excellent place to check your math. If the sum of the deviations of each score from the mean does not equal zero (or a number very, very close to zero in situations when you are rounding decimal places) then you have made a mathematical error either in your subtractions or your calculation of the mean.
Then, subtract the mean from every number to get the list of deviations. Create a list of these numbers. It’s OK to get negative numbers here.
Next, square the resulting list of numbers (read: multiply them with themselves) to get the squared deviation.
Add up all of the resulting squares to get their total sum.
To get the standard deviation, Divide your result by one less than the number of items in the list and just take the square root of the resulting number.
It is important for you to note that the last step in the calculation of the variance that I have described requires you to reduce the sample size by 1. This is done because we are using a sample rather than the entire population of officers. If our data is the actual entire population we would not subtract 1 from N. We would simply divide by the size of the entire population.. In this example the eight observations of the age of police officers is a sample from the total population of police officers in this jurisdiction. By subtracting 1 from the sample size (N – 1) we are adjusting the final value of the variance (s2) resulting in a value that is larger than if we were divide by N. When using sample data it is better to overstate the measure of dispersion than to understate it.
STANDARD DEVIATION – s
The standard deviation (s) is very simple to calculate.
In our example with the sample of 8 police officers and their Age, the standard deviation is:
S = √383.878/7 s = √54.84 s = 7.41years
The standard deviation has another very important advantage over the other measures of dispersion in that we are able to use the standard deviation to estimate the number of variable values within certain areas under the curve representative of those values.
Using the Standard Deviation
I have posted a pdf file under the “notes” link that outlines the areas falling under/within the normal curve/bell curve (or normal distribution). Please refer to that graphical display for the remainder of this discussion.
Our calculated mean is 28.375 years. When using the normal curve/bell curve (or normal distribution) to represent our variable we would place the mean, 28.375 years at the center of the distribution above X bar.
The numbers that correspond to the “-1s” and “+1s” are 20.965 (mean – standard deviation) and 35.785 (mean + standard deviation) respectively. These numbers are calculated by adding one standard deviation unit (7.41 years) to the mean of 28.375 years and subtracting one standard deviation unit (7.41 years) from the mean of 28.375 years. This represents the range of ages between which we would expect to find approximately 68.26% of the total population of police officers in this jurisdiction. We would expect approximately 34.13% to have an age between 20.965 years and 28.375 years. Similarly, we would expect approximately 34.13% to have an age between 28.375 years and 35.785 years.
The numbers that correspond to the “-2s” and “+2s” are 13.555 and 43.195 respectively. These numbers are calculated by adding two standard deviation units (7.41 years x 2 = 14.82 years) to the mean of 28.375 years and subtracting two standard deviation units (7.41 years x 2 = 14.82 years) from the mean of 28.375 years. This represents the range of ages between which we would expect to find approximately 95.44% of the total population of police officers in this jurisdiction. We would expect approximately 47.72% to have an age between 13.55 years and 28.375 years. Similarly, we would expect approximately 47.72% to have an age between 28.375 years and 43.195 years.
The numbers that correspond to the “-3s” and “+3s” are 6.145 and 50.605 respectively. These numbers are calculated by adding three standard deviation units (7.41 years x 3 = 22.23 years) to the mean of 28.375 years and subtracting three standard deviation units (7.41 years x 3 = 22.23 years) from the mean of 28.375 years. This represents the range of ages between which we would expect to find approximately 99.74% of the total population of police officers in this jurisdiction. We would expect approximately 49.87% to have an age between 6.145 and 28.375 years. Similarly, we would expect 49.87% to have an age between 28.375 years and 50.605 years.
More examples for practice:
STEP 1: Find the Mean for the distribution.
9 Mean= ΣX/N
8 = 30/6 = 5
ΣX = 30
STEP 2: Subtract the Mean from each raw score to get the DEVIATION.
X (X- Mean)-Deviation
9 (9-5) +4
8 (8-5) +3
6 (6-5) +1
4 (4-5) -1
2 (2-5) -3
1 (1-5) -4
STEP 3: Square each deviation before adding the SQUARED DEVIATIONS TOGETHER.
X (X- Mean)-Deviation (X- Mean) ² – Squared Deviation
9 (9-5) +4 16
8 (8-5) +3 9
6 (6-5) +1 1
4 (4-5) -1 1
2 (2-5) -3 9
1 (1-5) -4 16
Σ(X- Mean) ² = 52
STEP 4: Divide by N-1 and get the SQUARE ROOT OF THE RESULT FOR THE STANDARD DEVIATION.
S= √52/5 s= √10.4 s= 3.22
The following data represent the number of crime calls at “Hot Spots” in a year. Calculate and interpret the standard deviation of crime calls at these hot spots.
Hot Spot Number # of Calls (X- Mean)-Deviation (X- Mean) ² -Squared Deviation
1 2 -19.5 380.25
2 9 -12.5 156.25
3 11 -10.5 110.25
4 13 -8.5 72.25
5 20 -1.5 2.25
6 20 -1.5 2.25
7 20 -1.5 2.25
8 24 2.5 6.25
9 27 5.5 30.25
10 29 7.5 56.25
11 31 9.5 90.25
12 52 30.5 930.25
Σ=258 Σ=0 Σ= 1839
Mean = 258/12= 21.5 crime calls
s= √1839/11 s= √167.181 s= 12.93 crime calls
Why Work with Us
Top Quality and Well-Researched Papers
We always make sure that writers follow all your instructions precisely. You can choose your academic level: high school, college/university or professional, and we will assign a writer who has a respective degree.
Professional and Experienced Academic Writers
We have a team of professional writers with experience in academic and business writing. Many are native speakers and able to perform any task for which you need help.
Free Unlimited Revisions
If you think we missed something, send your order for a free revision. You have 10 days to submit the order for review after you have received the final document. You can do this yourself after logging into your personal account or by contacting our support.
Prompt Delivery and 100% Money-Back-Guarantee
All papers are always delivered on time. In case we need more time to master your paper, we may contact you regarding the deadline extension. In case you cannot provide us with more time, a 100% refund is guaranteed.
Original & Confidential
We use several writing tools checks to ensure that all documents you receive are free from plagiarism. Our editors carefully review all quotations in the text. We also promise maximum confidentiality in all of our services.
24/7 Customer Support
Our support agents are available 24 hours a day 7 days a week and committed to providing you with the best customer experience. Get in touch whenever you need any assistance.
Try it now!
How it works?
Follow these simple steps to get your paper done
Place your order
Fill in the order form and provide all details of your assignment.
Proceed with the payment
Choose the payment system that suits you most.
Receive the final file
Once your paper is ready, we will email it to you.
No need to work on your paper at night. Sleep tight, we will cover your back. We offer all kinds of writing services.
No matter what kind of academic paper you need and how urgent you need it, you are welcome to choose your academic level and the type of your paper at an affordable price. We take care of all your paper needs and give a 24/7 customer care support system.
Admission Essays & Business Writing Help
An admission essay is an essay or other written statement by a candidate, often a potential student enrolling in a college, university, or graduate school. You can be rest assurred that through our service we will write the best admission essay for you.
Our academic writers and editors make the necessary changes to your paper so that it is polished. We also format your document by correctly quoting the sources and creating reference lists in the formats APA, Harvard, MLA, Chicago / Turabian.
If you think your paper could be improved, you can request a review. In this case, your paper will be checked by the writer or assigned to an editor. You can use this option as many times as you see fit. This is free because we want you to be completely satisfied with the service offered.