# 2.7 Measures of the spread of the data  (Page 4/25)

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• For the following problems, recall that value = mean + (#ofSTDEVs)(standard deviation) . Verify the mean and standard deviation or a calculator or computer.
• For a sample: x = $\overline{x}$ + (#ofSTDEVs)( s )
• For a population: x = μ + (#ofSTDEVs)( σ )
• For this example, use x = $\overline{x}$ + (#ofSTDEVs)( s ) because the data is from a sample

1. Verify the mean and standard deviation on your calculator or computer.
2. Find the value that is one standard deviation above the mean. Find ( $\overline{x}$ + 1s).
3. Find the value that is two standard deviations below the mean. Find ( $\overline{x}$ – 2s).
4. Find the values that are 1.5 standard deviations from (below and above) the mean.
• Clear lists L1 and L2. Press STAT 4:ClrList. Enter 2nd 1 for L1, the comma (,), and 2nd 2 for L2.
• Enter data into the list editor. Press STAT 1:EDIT. If necessary, clear the lists by arrowing up into the name. Press CLEAR and arrow down.
• Put the data values (9, 9.5, 10, 10.5, 11, 11.5) into list L1 and the frequencies (1, 2, 4, 4, 6, 3) into list L2. Use the arrow keys to move around.
• Press STAT and arrow to CALC. Press 1:1-VarStats and enter L1 (2nd 1), L2 (2nd 2). Do not forget the comma. Press ENTER.
• $\overline{x}$ = 10.525
• Use Sx because this is sample data (not a population): Sx=0.715891
1. ( $\overline{x}$ + 1s) = 10.53 + (1)(0.72) = 11.25
2. ( $\overline{x}$ – 2 s ) = 10.53 – (2)(0.72) = 9.09
• ( $\overline{x}$ – 1.5 s ) = 10.53 – (1.5)(0.72) = 9.45
• ( $\overline{x}$ + 1.5 s ) = 10.53 + (1.5)(0.72) = 11.61

## Try it

On a baseball team, the ages of each of the players are as follows:

21; 21; 22; 23; 24; 24; 25; 25; 28; 29; 29; 31; 32; 33; 33; 34; 35; 36; 36; 36; 36; 38; 38; 38; 40

Use your calculator or computer to find the mean and standard deviation. Then find the value that is two standard deviations above the mean.

μ = 30.68

s = 6.09
( $\overline{x}$ + 2 s ) = 30.68 + (2)(6.09) = 42.86.

## Explanation of the standard deviation calculation shown in the table

The deviations show how spread out the data are about the mean. The data value 11.5 is farther from the mean than is the data value 11 which is indicated by the deviations 0.97 and 0.47. A positive deviation occurs when the data value is greater than the mean, whereas a negative deviation occurs when the data value is less than the mean. The deviation is –1.525 for the data value nine. If you add the deviations, the sum is always zero . (For [link] , there are n = 20 deviations.) So you cannot simply add the deviations to get the spread of the data. By squaring the deviations, you make them positive numbers, and the sum will also be positive. The variance, then, is the average squared deviation.

The variance is a squared measure and does not have the same units as the data. Taking the square root solves the problem. The standard deviation measures the spread in the same units as the data.

Notice that instead of dividing by n = 20, the calculation divided by n – 1 = 20 – 1 = 19 because the data is a sample. For the sample variance, we divide by the sample size minus one ( n – 1). Why not divide by n ? The answer has to do with the population variance. The sample variance is an estimate of the population variance. Based on the theoretical mathematics that lies behind these calculations, dividing by ( n – 1) gives a better estimate of the population variance.

what is standard deviation?
It is the measure of the variation of certain values from the Mean (Center) of a frequency distribution of sample values for a particular Variable.
Dominic
Yeah....the simplest one
IRFAN
what is the number of x
10
Elicia
Javed Arif
Jawed
how will you know if a group of data set is a sample or population
population is the whole set and the sample is the subset of population.
umair
if the data set is drawn out of a larger set it is a sample and if it is itself the whole complete set it can be treated as population.
Bhavika
hello everyone if I have the data set which contains measurements of each part during 10 years, may I say that it's the population or it's still a sample because it doesn't contain my measurements in the future? thanks
Alexander
Pls I hv a problem on t test is there anyone who can help?
Peggy
Dominic
Bhavika is right
Dominic
what is the problem peggy?
Bhavika
hi
Sandeep
Hello
hi
Bhavika
hii Bhavika
Dar
Hi eny population has a special definition. if that data set had all of characteristics of definition, that is population. otherwise that is a sample
Hoshyar
three coins are tossed. find the probability of no head
three coins are tossed consecutively or what ?
umair
umair
or .125 is the probability of getting no head when 3 coins are tossed
umair
🤣🤣🤣
Simone
what is two tailed test
if the diameter will be greater than 3 cm then the bullet will not fit in the barrel of the gun so you are bothered for both the sides.
umair
in this test you are worried on both the ends
umair
lets say you are designing a bullet for thw gun od diameter equals 3cm.if the diameter of the bullet is less than 3 cm then you wont be able to shoot it
umair
In order to apply weddles rule for numerical integration what is minimum number of ordinates
excuse me?
Gabriel
why?
didn't understand the question though.
Gabriel
which question? ?
We have rules of numerical integration like Trapezoidal rule, Simpson's 1/3 and 3/8 rules, Boole's rule and Weddle rule for n =1,2,3,4 and 6 but for n=5?
John
geometric mean of two numbers 4 and 16 is:
10
umair
really
iphone
quartile deviation of 8 8 8 is:
iphone
sorry 8 is the geometric mean of 4,16
umair
quartile deviation of 8 8 8 is
iphone
can you please expalin the whole question ?
umair
mcq
iphone
h
iphone
can you please post the picture of that ?
umair
how
iphone
hello
John
10 now
John
how to find out the value
can you be more specific ?
umair
yes
KrishnaReddy
what is the difference between inferential and descriptive statistics
descriptive statistics gives you the result on the the data like you can calculate various things like variance,mean,median etc. however, inferential stats is involved in prediction of future trends using the previous stored data.
umair
if you need more help i am up for the help.
umair
Thanks a lot
Anjali
Inferential Statistics involves drawing conclusions on a population based on analysis of a sample. Descriptive statistics summarises or describes your current data as numerical calculations or graphs.
fred
my pleasure😊. Helping others offers me satisfaction 😊
umair
for poisson distribution mean............variance.
both are equal to mu
Faizan
mean=variance
Faizan
what is a variable
something that changes
Festus
why we only calculate 4 moment of mean? asked in papers.
why we only 4 moment of mean ? asked in BA exam
Faizan
Hello, can you please share the possible questions that are likely to be examined under the topic: regression and correlation analysis.
Refiloe
for normal distribution mean is 2 & variance is 4 find mu 4?
repeat quastion again
Yusuf
find mu 4. it can be wrong but want to prove how.
Faizan
for a normal distribution if mu 4 is 12 then find mu 3?
Question hi wrong ha
Tahir
ye BA mcqs me aya he teen he. 2dafa aya he
Faizan
if X is normally distributed. (n,b). then its mean deviation is?
Faizan
The answer is zero, because all odd ordered central moments of a normal distribution are Zero.
nikita
which question is zero
Faizan
sorry it is (5,16) in place of (n,b)
Faizan
I got. thanks. it is zero.
Faizan
a random variable having binomial distribution is?
Bokaho