# 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.

the probability of an event occuring is defined as?
The probability of an even occurring is expected event÷ event being cancelled or event occurring / event not occurring
Gokuna
what is simple bar chat
Simple Bar Chart is a Diagram which shows the data values in form of horizontal bars. It shows categories along y-axis and values along x-axis. The x-axis displays above the bars and y-axis displays on left of the bars with the bars extending to the right side according to their values.
statistics is percentage only
the first word is chance for that we use percentages
it is not at all that statistics is a percentage only
Shambhavi
I need more examples
how to calculate sample needed
mole of sample/mole ratio or Va Vb
Gokuna
how to I solve for arithmetic mean
Yeah. for you to say.
James
yes
niharu
how do I solve for arithmetic mean
niharu
add all the data and divide by the number of data sets. For example, if test scores were 70, 60, 70, 80 the total is 280 and the total data sets referred to as N is 4. Therfore the mean or arthritmatic average is 70. I hope this helps.
Jim
*Tan A - Tan B = sin(A-B)/CosA CosB ... *2sinQ/Cos 3Q = tan 3Q - tan Q
standard error of sample
what is subjective probability
how to calculate the Steadman rank correlation
David
what is sampling? i want to know about the definition of sampling.
what is sample...?
In terms of Statistics or Research , It is a subset of population for measurement.
Da
can you solve this problem
yes
Harry
which problem
Larwubah
what is the meaning of correlation ratio?
Nayeem
in 2018,walewale hospital recorded 2500cases of infection it was seen that out of this number 350 cases are rti 150 were bronchitis 300 cases were otitis media the rest were peptic ulcer cases calculate proportion of peptic ulcer and percentage of bronchitis
what is statistics
peter
yo
Kailesh
what is the frequency
Frequency is the number of all object which is comes from population or sample size
Faiqa
Denoted by f
Faiqa
number of all objects?
Amir
frequency is the rate of occurrence of an object
Leek
Explain nominal and ordinal variables
Oyinlola
nominal variables are those variable which is used to “name,” a series of values.
Amir
while  ordinal scales provide good information about the order of choices,for example in a customer satisfaction survey.
Amir
what is the difference between Mean and Varience?
Amir
Sum of total object, divided by number of object is called mean
Faiqa
variance?
Amir
faiqa U didn't clear me.Sorry
Amir
what is df in statistics
Oyinlola
degre of freedom
Amir