THE PIONEER BLOG

The Standard Deviation Formula

This is the formula for the standard deviation sx of the set of data xi.

How to Calculate a Standard Deviation

So you have a set of data and want to know how spread out it is. There are a few ways to do this, but one of the most useful is called the standard deviation. Your calculator may have a built in standard deviation button, which typically has a sx on it. But sometimes it’s nice to know what your calculator is doing behind the scenes. The steps below actually break down the formula for standard deviation into a process. If you're ever asked to do a problem like this on a test, know that sometimes it’s easier to remember a step by step process rather than memorizing a formula.

The Process

Calculate the mean of your data set.
Subtract the mean from each of the data values and list the differences.
Square each of the differences from the previous step and make a list of the squares.
- In other words, multiply each number by itself.
- Be careful with negatives. A negative times a negative makes a positive.
Add the squares from the previous step together.
Subtract one from the number of data values you started with.
Divide the sum from step four by the number from step five.
Take the square root of the number from the previous step. This is the standard deviation.
- You may need to use a basic calculator to find the square root.
- Be sure to use significant figures when rounding your answer.

A Worked Example

Suppose you're given the data set 1,2,2,4,6. Work through each of the steps to find the standard deviation.

Calculate the mean of your data set.
The the mean of the data is (1+2+2+4+6)/5 = 15/5 = 3.
Subtract the mean from each of the data values and list the differences.
Subtract 3 from each of the values 1,2,2,4,6
1-3 = -2
2-3 = -1
2-3 = -1
4-3 = 1
6-3 = 3
Your list of differences is -2,-1,-1,1,3
Square each of the differences from the previous step and make a list of the squares.
You need to square each of the numbers -2,-1,-1,1,3
Your list of differences is -2,-1,-1,1,3
(-2)2 = 4
(-1)2=1
(-1)2=1
12=1
32=9
Your list of squares is 4,1,1,1,9
Add the squares from the previous step together.
You need to add 4+1+1+1+9=16
Subtract one from the number of data values you started with.
You began this process (it may seem like awhile ago) with five data values. One less than this is 5-1 = 4.
Divide the sum from step four by the number from step five.
The sum was 16, and the number from the previous step was 4. You divide these two numbers 16/4 = 4.
Take the square root of the number from the previous step. This is the standard deviation.
Your standard deviation is the square root of 4, which is 2.

Tip: It’s sometimes helpful to keep everything organized in a table, like the one shown below.

Data	Data-Mean	(Data-Mean)2
1	-2	4
2	-1	1
2	-1	1
4	1	1
6	3	9

How to Make a Stem Plot

When I’m done grading an exam, I like to see how well the class did as a whole. Most of the time there’s not a calculator handy, so I don’t start by calculating the mean or median

. Instead it’s helpful to see how the scores are distributed. Do they resemble a bell curve? Are the scores bimodal? One type of graph that displays these features of the data is called a stem and leaf plot, or stemplot. Despite the name, there is no flora or foliage involved. Instead the idea is that stems form one part of a number, and leaves are the rest of that number.

To make my stemplot, each score is broken into two pieces, the stem and leaf. In this particular case the tens digits are stems, and the ones digits form the leaves. The resulting stemplot produces a distribution of the data similar to a histogram, but all of the data values are retained in a compact form. Features of the students’ performance can be easily seen from the shape of the stem and leaf plot.

Suppose that my class had the following test scores: 84, 65, 78, 75, 89, 90, 88, 83, 72, 91, 90 and we wanted to see at a glance what features were present in the data. We rewrite the list of scores in order and then utilize a stem and leaf plot. The stems are 6, 7, 8, 9, corresponding to the tens place of the data. This is listed in a vertical column. The ones digit of each score is written in a horizontal row to the right of each stem:

9| 0 0 1

8| 3 4 8 9

7| 2 5 8

6| 2

The data can be easily read from the stemplot. For example, the top row contains the values 90, 90 and 91.

What’s the Stem and What’s the Leaf?

With test scores, as well as other data, that range between 0 and 100 points the above strategy works for choosing stems and leaves. But how do we know what to pick for a stem and a leaf with other sets of data? For data with more than two digits there are some options. It all comes down to how the data values are distributed.

If we wanted to make a stem and leaf plot for the data 100, 105, 110, 120, 124, 126, 130, 131, 132 what would we choose for a stem? If we say the highest place value, that is the hundreds digit is the stem, our resulting stemplot is not very helpful because none of the values are separated from any of the others:

1|00 05 10 20 24 26 30 31 32

Instead we could try to make the stem the first two digits of the data. The resulting stem and leaf plot does a better job at depicting the data:

13| 0 1 2

12| 0 4 6

11| 0

10| 0 5

Expanding and Condensing

The two stemplots in the previous section show how versatile stem and leaf plots are. They can be expanded or condensed by changing the form of the stem. One strategy for expanding a stemplot is to evenly split a stem into equally sized pieces. Consider the stemplot:

9| 0 0 1

8| 3 4 8 9

7| 2 5 8

6| 2

We expand this stem and leaf plot by splitting each stem into two. This results in two stems for each tens digit. The data with 0-4 in the ones place value are separated from those with digits 5-9 in the ones place:

9| 0 0 1

8| 8 9

8| 3 4

7| 5 8

7| 2

6| 2

Here the 6 with no numbers to the right shows that there are no data values from 65 to 69.

Binomial Table For n = 7 to n = 9

The binomial distribution gives the probability of r successes in an experiment with a total of nindependent trials, each having probability of success p. The tables below are for n = 7, 8, and 9.

Each entry in the table is calculated by the formula C(n, r)pr(1 - p)n - r where C(n, r) is the formula for combinations.

Other binomial distribution tables: n = 2 to 6, n = 10 to 11

Tables for n = 7 to n = 9

n = 7

.01

.05

.10

.15

.20

.25

.30

.35

.40

.45

.50

.55

.60

.65

.70

.75

.80

.85

.90

.95

.932

.698

.478

.321

.210

.133

.082

.049

.028

.015

.008

.004

.002

.001

.000

.066

.257

.372

.396

.367

.311

.247

.185

.131

.087

.055

.032

.017

.008

.004

.001

.000

.002

.041

.124

.210

.275

.311

.318

.299

.261

.214

.164

.117

.077

.047

.025

.012

.004

.001

.000

.004

.023

.062

.115

.173

.227

.268

.290

.292

.273

.239

.194

.144

.097

.058

.029

.011

.003

.000

.003

.011

.029

.058

.097

.144

.194

.239

.273

.292

.290

;268

.227

.173

.115

.062

.023

.004

.000

.001

.004

.012

.025

.047

.077

.117

.164

.214

.261

.299

.318

.311

.275

.210

.124

.041

.000

.001

.004

.008

.017

.032

.055

.087

.131

.185

.247

.311

.367

.396

.372

.257

.000

.001

.002

.004

.008

.015

.028

.049

.082

.133

.210

.321

.478

.698

n = 8

.01

.05

.10

.15

.20

.25

.30

.35

.40

.45

.50

.55

.60

.65

.70

.75

.80

.85

.90

.95

.923

.663

.430

.272

.168

.100

.058

.032

.017

.008

.004

.002

.001

.000

.075

.279

.383

.385

.336

.267

.198

.137

.090

.055

.031

.016

.008

.003

.001

.000

.003

.051

.149

.238

.294

.311

.296

.259

.209

.157

.109

.070

.041

.022

.010

.004

.001

.000

.005

.033

.084

.147

.208

.254

.279

.257

.219

.172

.124

.081

.047

.023

.009

.003

.000

.005

:018

.046

.087

.136

.188

.232

.263

.273

.263

.232

.188

.136

.087

.046

.018

.005

.000

.003

.009

.023

.047

.081

.124

.172

.219

.257

.279

.254

.208

.147

.084

.033

.005

.000

.001

.004

.010

.022

.041

.070

.109

.157

.209

.259

.296

.311

.294

.238

.149

.051

.000

.001

.003

.008

.016

.031

.055

.090

.137

.198

.267

.336

.385

.383

.279

.000

000

.000

.001

.002

.004

.008

.017

.032

.058

.100

.168

.272

.430

.663

n = 9

.01

.05

.10

.15

.20

.25

.30

.35

.40

.45

.50

.55

.60

.65

.70

.75

.80

.85

.90

.95

.914

.630

.387

.232

.134

.075

.040

.021

.010

.005

.002

.001

.000

.083

.299

.387

.368

.302

.225

.156

.100

.060

.034

.018

.008

.004

.001

.000

.003

.063

.172

.260

.302

.300

.267

.216

.161

.111

.070

.041

.021

.010

.004

.001

.000

.008

.045

.107

.176

.234

.267

.272

.251

.212

.164

.116

.074

.042

.021

.009

.003

.001

.000

.001

.007

.028

.066

.117

.172

.219

.251

.260

.246

.213

.167

.118

.074

.039

.017

.005

.001

.000

.001

.005

.017

.039

.074

.118

.167

.213

.246

.260

.251

.219

.172

.117

.066

.028

.007

.001

.000

.001

.003

.009

.021

.042

.074

.116

.164

.212

.251

.272

.267

.234

.176

.107

.045

.008

.000

.001

.004

.010

.021

.041

.070

.111

.161

.216

.267

.300

.302

.260

.172

.063

.000

.001

.004

.008

.018

.034

.060

.100

.156

.225

.302

.368

.387

.299

.000

.001

.002

.005

.010

.021

.040

.075

.134

.232

.387

.630

Combinations and Permutations How Can We Tell the Difference Between Them?

To figure out some probability problems, we must be able to count. Suppose we are given a total of n distinct objects and want to select r of them. This touches directly on an area of mathematics known as combinatorics, which is the study of counting. Two of the main ways to count are called permutations and combinations. These concepts are closely related to one another and easily confused.

How can we tell the difference between a combination and permutation? The key idea is that of order. A permutation pays attention to the order that we select our objects. The same set of objects, but taken in a different order will give us different permutations. With a combination we still select r objects from a total of n, but the order is no longer considered.

An Example of Permutations

To distinguish between these ideas, we will consider the following example: how many permutations are there of two letters from the set {a,b,c}?

Here we list all pairs of elements from the given set, all the while paying attention to the order. There are a total of six permutations. The list of all of these are: ab, ba, bc, cb, ac, ca. Note that as permutationsab and ba are different because in one case a was chosen first, and in the other a was chosen second.

An Example of Combinations

Now we will answer the following question: how many combinations are there of two letters from the set {a,b,c}?

Since we are dealing with combinations, we no longer care about the order. We can solve this problem by looking back at the permutations, and then eliminating those that include the same letters. As combinations, ab and ba are regarded as the same. Thus there are only three combinations: ab, ac, bc.

Formulas

For situations we encounter with larger sets it is too time consuming to list out all of the possible permutations or combinations and count the end result. Fortunately there are formulas that give us the number of permutations or combinations of n objects taken r at a time.

In these formulas we use the shorthand notation of n!, called n factorial. The factorial simply says to multiply all positive whole numbers less than or equal to n together. So, for instance, 4! = 4 x 3 x 2 x 1 = 24. By definition 0! = 1.

The number of permutations of n objects taken r at a time is given by the formula:

P(n,r) = n!/(n - r)!

The number of combinations of n objects taken r at a time is given by the formula:

C(n,r) = n!/[r!(n - r)!]

Formulas at Work

To see the formulas at work, let’s look at the initial example. The number of permutations of a set of three objects taken two at a time is given by P(3,2) = 3!/(3 - 2)! = 6/1 = 6. This matches exactly what we obtained by listing all of the permutations.

The number of combinations of a set of three objects taken two at a time is given by:

C(3,2) = 3!/[2!(3-2)!] = 6/2 = 3. Again, this lines up exactly with what we saw before.

The formulas definitely save time when we are asked to find the number of permutations of a larger set. For instance, how many permutations are there of a set of ten objects taken three at a time. It would take awhile to list all the permutations, but with the formulas we see that there would be:

P(10,3) = 10!/(10-3)! = 10!/7! = 10 x 9 x 8 = 720 permutations.

The Main Idea

What is the difference between permutations and combinations? The bottom line is that in counting situations that involve an order, permutations should be used. If the order is not important, then combinations should be utilized.

Binomial Probability Distribution Formula

The above formula indicates the probability of r successes out of n independent trials, where each success has probability p of occurring.

The formula can be thought of in the following way. The probability of r successes occuring, each with probability p is pr. This leaves n-r failures, and each failure has probability 1-p. We multiply all of these probabilities together because the trials are independent, and this gives us pr(1-p)n-r.

The factorial in the formula shows up because we need to find the combination of ways that we can have r successes out of n trials.

Most calculations involving the binomial distribution will be carried out with software or tables. However, it is important to realize that the values in the table do not fall out of the sky. Only a few mathematical ideas from probability and combinatorics are needed to derive tables of the binomial probability distribution.

Binomial Table For n = 2 to n = 6

The binomial distribution gives the probability of r successes in an experiment with a total of nindependent trials, each having probability of success p. The tables below are for n = 2, 3, 4, 5 and 6.

Each entry in the table is calculated by the formula C(n, r)pr(1 - p)n - r where C(n, r) is the formula for combinations.

Other binomial distribution tables: n = 7 to 9, n = 10 to 11.

Tables for n=2 to n=6

n = 2

	p	.01	.05	.10	.15	.20	.25	.30	.35	.40	.45	.50	.55	.60	.65	.70	.75	.80	.85	.90	.95
r	0	.980	.902	.810	.723	.640	.563	.490	.423	.360	.303	.250	.203	.160	.123	.090	.063	.040	.023	.010	.002
	1	.020	.095	.180	.255	.320	.375	.420	.455	.480	.495	.500	.495	.480	.455	.420	.375	.320	.255	.180	.095
	2	.000	.002	.010	.023	.040	.063	.090	.123	.160	.203	.250	.303	.360	.423	.490	.563	.640	.723	.810	.902

n = 3

	p	.01	.05	.10	.15	.20	.25	.30	.35	.40	.45	.50	.55	.60	.65	.70	.75	.80	.85	.90	.95
r	0	.970	.857	.729	.614	.512	.422	.343	.275	.216	.166	.125	.091	.064	.043	.027	.016	.008	.003	.001	.000
	1	.029	.135	.243	.325	.384	.422	.441	.444	.432	.408	.375	.334	.288	.239	.189	.141	.096	.057	.027	.007
	2	.000	.007	.027	.057	.096	.141	.189	.239	.288	.334	.375	.408	.432	.444	.441	.422	.384	.325	.243	.135
	3	.000	.000	.001	.003	.008	.016	.027	.043	.064	.091	.125	.166	.216	.275	.343	.422	.512	.614	.729	.857

n = 4

	p	.01	.05	.10	.15	.20	.25	.30	.35	.40	.45	.50	.55	.60	.65	.70	.75	.80	.85	.90	.95
r	0	.961	.815	.656	.522	.410	.316	.240	.179	.130	.092	.062	.041	.026	.015	.008	.004	.002	.001	.000	.000
	1	.039	.171	.292	.368	.410	.422	.412	.384	.346	.300	.250	.200	.154	.112	.076	.047	.026	.011	.004	.000
	2	.001	.014	.049	.098	.154	.211	.265	.311	.346	.368	.375	.368	.346	.311	.265	.211	.154	.098	.049	.014
	3	.000	.000	.004	.011	.026	.047	.076	.112	.154	.200	.250	.300	.346	.384	.412	.422	.410	.368	.292	.171
	4	.000	.000	.000	.001	.002	.004	.008	.015	.026	.041	.062	.092	.130	.179	.240	.316	.410	.522	.656	.815

n = 5

	p	.01	.05	.10	.15	.20	.25	.30	.35	.40	.45	.50	.55	.60	.65	.70	.75	.80	.85	.90	.95
r	0	.951	.774	.590	.444	.328	.237	.168	.116	.078	.050	.031	.019	.010	.005	.002	.001	.000	.000	.000	.000
	1	.048	.204	.328	.392	.410	.396	.360	.312	.259	.206	.156	.113	.077	.049	.028	.015	.006	.002	.000	.000
	2	.001	.021	.073	.138	.205	.264	.309	.336	.346	.337	.312	.276	.230	.181	.132	.088	.051	.024	.008	.001
	3	.000	.001	.008	.024	.051	.088	.132	.181	.230	.276	.312	.337	.346	.336	.309	.264	.205	.138	.073	.021
	4	.000	.000	.000	.002	.006	.015	.028	.049	.077	.113	.156	.206	.259	.312	.360	.396	.410	.392	.328	.204
	5	.000	.000	.000	.000	.000	.001	.002	.005	.010	.019	.031	.050	.078	.116	.168	.237	.328	.444	.590	.774

n = 6

	p	.01	.05	.10	.15	.20	.25	.30	.35	.40	.45	.50	.55	.60	.65	.70	.75	.80	.85	.90	.95
r	0	.941	.735	.531	.377	.262	.178	.118	.075	.047	.028	.016	.008	.004	.002	.001	.000	.000	.000	.000	.000
	1	.057	.232	.354	.399	.393	.356	.303	.244	.187	.136	.094	.061	.037	.020	.010	.004	.002	.000	.000	.000
	2	.001	.031	.098	.176	.246	.297	.324	.328	.311	.278	.234	.186	.138	.095	.060	.033	.015	.006	.001	.000
	3	.000	.002	.015	.042	.082	.132	.185	.236	.276	.303	.312	.303	.276	.236	.185	.132	.082	.042	.015	.002
	4	.000	.000	.001	.006	.015	.033	.060	.095	.138	.186	.234	.278	.311	.328	.324	.297	.246	.176	.098	.031
	5	.000	.000	.000	.000	.002	.004	.010	.020	.037	.061	.094	.136	.187	.244	.303	.356	.393	.399	.354	.232
	6	.000	.000	.000	.000	.000	.000	.001	.002	.004	.008	.016	.028	.047	.075	.118	.178	.262	.377	.531	.735