MTH 110 – INTRODUCTION TO
STATISTICS
Session 4 – Measures of Dispersion, Variance and Standard Deviation Instructor: Manos Takas Email: m.takas@cityu.gr
Range Look at these two sets of data: 2, 3, 4, 5, 6 3,2,3,5,13 3,2,3,5,13 They both have a mean of 4. However, you can see that the first data set (21 is more spread out than the second data set. The mean doesn't tell you this. _ To represent the data more accurately, you need the mean plus a measure of the spread or dispersion of the data. One simple measure of dispersion is the range. The range of a set of data is the highest value minus the lowest value
In this case the range of set [1] is 6  2 = 4. The range of set [2] is 13 (3) = 16. The range is easy to calculate, but there are other measures of spread that are more useful.
1
24/10/2011
Variance and Standard Deviation The mean of the deviations from the mean squared is called the variance, usually written as σ 2
The variance is calculated as:
σ2
=
1 ∑ ( x − x )2 n
2
Sometimes it is easier to calculate a variance using the alternative formula:
σ
⎛∑x⎞ 1 = ∑ x2 − ⎜ ⎜ n ⎟ ⎟ n ⎝ ⎠
2
Standard Deviation The standard deviation σ is defined as the square root of the variance
⎛ ∑ x2 ⎞ 1 1 2 σ= ∑ x − ⎜ n ⎟ = n ∑ ( x − x )2 ⎜ ⎟ n ⎝ ⎠ 2
Standard Deviation for Frequency Distributions
σ2 =
1 ∑ n
⎛ ∑ fx ⎞ fx 2 − ⎜ ⎜ n ⎟ ⎟ ⎝ ⎠
2
Where f represents the frequency of the x observation and n=Σf
...spertion
MEASURES OF DISPERSION
I. Meaning and Types of Measures of DispersionMeasures of Dispersion ( Variation or Spread )
 is a measure used to describe the variation of a set of data
Measure of Dispersion Symbol for 
 Parameter Sample 
A. ABSOLUTE DISPERSION   
Range R R 
Average Deviation A.D. A.D. 
Variance δ2 s2 
Standard Deviation Δ s 
Others   
B. RELATIVE DISPERSION   
Coefficient of Variation C.V. c.v. 
Standard Score   
II. ABSOLUTE DISPERSION
A. RANGE
Meaning: Range is the difference between the highest and lowest values...
...201315
Measures of Dispersion – Comparison of Cargo shipped via Airline Vs Rail Sector for last five years
In statistics, dispersion (also called variability, scatter, or spread) denotes how stretched or squeezed is a distribution (theoretical or that underlying a statistical sample). Common examples of measures of statistical dispersion are the variance, standard deviation and interquartile rang..
Dispersion is contrasted with location or central tendency, and together they are the most used properties of distributions.
Measures of statistical dispersion
A measure of statistical dispersion is a nonnegative real number that is zero if all the data are the same and increases as the data become more diverse.
Most measures of dispersion have the same units as the quantity being measured. In other words, if the measurements are in meters or seconds, so is the measure of dispersion. Such measures of dispersion include:
Sample standard deviation
Interquartile range (IQR) or Interdecile range
Range
Mean difference
Median absolute deviation (MAD)
Average absolute deviation (or simply called average deviation)
Distance standard deviatio
These are frequently used (together with scale factors) as...
...skills to solve problems mentally. The marks scored by pupils generate statistics which are used by teachers to analyse a student’s performance and development of theories to explain the differences in performance.
The Standard 3 class is where the transition from junior to senior level occurs where teachers expect the transference of concrete to abstract thinking would have occurred.
A common theory by many primary school teachers is ‘Students perform better in Mathematics than Mental math. Mental math is something that has to be developed and involves critical thinking. Mental math requires quick thinking and the student must solve the problem in their minds whereas in regular mathematics, the problem can be solved visually. Therefore, teachers should take these factors into consideration while testing and marking students in these areas.’
In this study, the statistics of 30 students of a standard 3 class of San Fernando Boys’ Government School will be analysed to determine the truth of this theory.
DATA COLLECTION METHODS
Mathematics and mental mathematics marks of term 1 of the class of 2013 were obtained from a Standard 3 teacher of San Fernando Boys’ Government School.
The marks were divided into the form:
Mathematics Marks, x
Mental Mathematics marks, y
DATA PRESENTATION AND ANALYSIS
Sample number
Marks earned (x)
x²
1
70
4900
2
86
7396
3
49
2401
4
66
4356
5...
...
Contents
Question 1 3
Question 2a 5
Question 2b 6
Question 2c 7
Question 3a 8
Question 3b 8
Question 3c 10
Question 3d 11
Question 4 12
Question 5 14
References 15
Question 1
The sampling method that Mr. Kwok is using is Stratified Random Sampling Method. In this case study, Mr Kwok collected a random sample of 1000 flights and proportions of three routes in the sample. He divides them into different subgroups such as satisfaction, refreshments and departure time and then selects proportionally to highlight specific subgroup within the population. The reasons why Mr Kwok used this sampling method are that the cost per observation in the survey may be reduced and it also enables to increase the accuracy at a given cost.
TABLE 1: Data Summaries of Three Routes
Route 1
Route 2
Route 3
Normal(88.532,5.07943)
Normal(97.1033,5.04488)
Normal(107.15,5.15367)
Summary Statistics
Mean
88.532
Std Dev
5.0794269
Std Err Mean
0.2271589
Upper 95% Mean
88.978306
Lower 95% Mean
88.085694
N
500
Sum
44266
Summary Statistics
Mean
97.103333
Std Dev
5.0448811
Std Err Mean
0.2912663
Upper 95% Mean
97.676525
Lower 95% Mean
96.530142
N
300
Sum
29131
Summary Statistics
Mean
107.15
Std Dev
5.1536687
Std Err Mean
0.3644194
Upper 95% Mean
107.86862...
...R.A. Fisher and George W. Snedecor)(2) which arises in the testing of whether two observed samples have the same variance. (1)
Note that three of the most important distributions (namely the normal distribution, the t distribution, and the chisquare distribution) may be seen as special cases of the F distribution: (3)
Example: We want to measure the monthly sales volume from Microsoft and Apple. We collect data for a year ( 12 months). We calculate the variance for both and define the “degrees of freedom’ (n1= 11) and then we can build the Fdistribution.
F statistic ():
Defined as the ratio of the dispersions of the two distributions, in other words it is the value calculated by the ratio of two sample variances . F always >=1.
The F statistic can test the null hypothesis: (1) that the two sample variances are from normal populations with a common variance; (2) that two population means are equal; (3) that no connection exists between the dependent variable and all or some of the independent variables.

Where and be independent variates distributed as chisquared with and degrees of freedom.
Example: We want to measure the monthly sales volume from Microsoft and Apple. We collect data for a year ( 12 months). We calculate the variance for both and define the “degrees of freedom’ (n1= 11) . Then we calculate F= (V² (M) /m)/ V²(A)/a, where V(M) variance for Microsoft, V(A) variance...
...perceptions, and their spending behaviors. In particular, part 2 examines how conditional probabilities related to spending behavior might vary, depending on the gender of the respondent.
Based on the relative frequencies for responses to each variable, determine the probability that a randomly selected respondent from:
a. [variable 4] spends at least $15 during a trip to Springdale Mall.
b. [variable 5] spends at least $15 during a trip to Downtown.
d. [variable 6] spends at least $15 during a trip to West Mall.
Comparing the preceding probabilities, which areas seem strongest and weakest in terms of the amount of money a shopper spends during a typical shopping visit?
e. [variable 11] feels that Springdale Mall has the highestquality goods.
f. [variable 11] feels that Downtown has the highestquality goods.
g. [variable 11] feels that West Mall has the highestquality goods.
Comparing the preceding probabilities, which areas are strongest and weakest in terms of the quality of goods offered?
Set up a contingency table for the appropriate variables given, then determine the following probabilities:
a. [variables 4 and 26] Given that the random respondent is a female, what is the probability that she spends at least $15 during a trip to Springdale Mall? Is a male more likely or less likely than a female to spend at least $15 during a visit to this area?
b. [variables 5 and 26] Given that the random respondent is...
...is reached
C. RNA polymerase can only add to the 3’ end
D. Transcription occurs in the 5’ to 3’ direction
E. An RNA transcript is the end result
F. All three types of RNA are transcribed from DNA
Name 3 classes of RNA and their function.
Ribosomal RNA, which is the site of protein synthesis.
Transfer RNA, which transports the correct amino acids to ribosomes and pairs them up.
Messenger RNA, which is the genetic blue print for making proteins.
2. What is the function of RNA polymerase and the promoter?
It adds complimentary ribonucleotides and a promoter is the base sequence in the DNA that signals the start of a gene.
3. List 3 ways RNA is modified.
Addition of a 5’ cap, Addition of a 3’ poly A tail, and introns being removed.
4. What is the genetic code? What does it mean to state the genetic code is
redundant and practically universal?
Most of the code is the same for all organisms.
5. What is the difference between an anticodon and a codon?
A codon is a sequence of three nucleotides that together form a unit of genetic code of DNA or RNA. An anticodon is a sequence of three nucleotides that together form a unit of genetic code in the transfer RNA.
6. Describe the stages of translation. Include initiation, elongation and termination in
your explanation.
The initiator tRNA binds to small ribosomal subunits. mRNA passes through, tRNAs deliver amino acids to the ribosomal binding site in the elongation, a stop codon in the mRNA moves onto...
...Trajico, Maria Liticia D.
BSEd IIIA2
REFLECTION
The first thing that puffs in my mind when I heard the word STATISTIC is that it was a very hard subject because it is another branch of mathematics that will make my head or brain bleed of thinking of how I will handle it. I have learned that statistic is a branch of mathematics concerned with the study of information that is expressed in numbers, for example information about the number of times something happens. As I examined on what the statement says, the phrase “number of times something happens” really caught my attention because my subconscious says “here we go again the nonstop solving, analyzing of problems” and I was right. This course of basic statistic has provided me with the analytical skills to crunch numerical data and to make inference from it. At first I thought that I will be alright all along with this subject but it seems that just some part of it maybe it is because I don’t pay much of my attention to it but I have learned many things. I have learned my lesson.
During our every session in this subject before having our midterm examination I really had hard and bad times in coping up with this subject. When we have our very first quiz I thought that I would fail it but it did not happen but after that, my next quizzes I have taken I failed. I was always feeling down when in every quiz I failed because even though I don’t...