1. Consider a sample with data values of 27, 25, 20, 15, 30, 34, 28, and 25. a) Compute the mean, median, and mode.
b) Compute the 20th, 65th, and 75th percentiles.
c) Compute the range, interquartile range, variance, and standard deviation.

Answers:
Data values: 15, 20, 25, 25, 27, 28, 30, 34
a) Mean: [pic]= ∑xi/n = (15+20+25+25+27+28+30+34) / 8 = 204 / 8 = 25.5 Median: Even number, so median is = (25+27)/2 = 26
Mode: Most frequent number = 25

c) Range: Largest data value – smallest data value = 34-15 = 19 Interquartile range: 3rd Quartile – 1st Quartile = 6 –((25/100)8) = 6-2 = 4 Variance: [pic]= (204-25.5)/8-1 = 25.5
Standard Deviation: [pic]= [pic]= 5.05

2. Consider a sample with a mean of 500 and a standard deviation of 100. What are the z-scores for the following data values: 650, 500, and 280?

Answers:
a) Data value 650: z-scores = [pic] = (650-500)/100 = 1.5 b) Data value 500: (500-500)/100 = 0
c) Data value 280: (280-500)/100 = -2.2

3. Consider a sample with a mean of 30 and a standard deviation of 5. Use Chebyshev’s theorem to determine the percentage of the data within each of the following ranges. a) 20 to 40
b) 15 to 45
c) 18 to 42

Answers:
Standard deviation (s) = 5
Mean ([pic]) = 30
a) z1 = (20-30)/5 = -2
z2 = (40-30)/5 = 2
(1-1/z²) = (1-1/2²) = 0.75 = 75%

...Mean, Median, Mode, and Range
Mean, median, and mode are three kinds of "averages". There are many "averages" in statistics, but these are, I think, the three most common, and are certainly the three you are most likely to encounter in your pre-statistics courses, if the topic comes up at all.
The "mean" is the "average" you're used to, where you add up all the numbers and then divide by thenumber of numbers. The "median" is the "middle" value in the list of numbers. To find the median, your numbers have to be listed in numerical order, so you may have to rewrite your list first. The "mode" is the value that occurs most often. If no number is repeated, then there is no mode for the list.
The "range" is just the difference between the largest and smallest values.
Find the mean, median, mode, and range for the following list of values:
13, 18, 13, 14, 13, 16, 14, 21, 13
The mean is the usual average, so:
(13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13) ÷ 9 = 15
Note that the mean isn't a value from the original list. This is a common result. You should not assume that your mean will be one of your original numbers.
The median is the middle value, so I'll have to rewrite the list...

...referred to as the arithmetic mean or simply the mean. The mean is a measure of the center of the data. Average and mean are used interchangeably to label the result of the sum of all measurements divided by the number of measurements. In mathematical notation the formula for calculating the sample mean is given below.
If the value given represents the mean of all values in a population it is denoted .
When the data are from a sample, the calculated value, in this case the mean, is referred to as a statistic. When the data represent the entire population, the value is referred to as a parameter.
The primary goal of this course is to learn techniques for which we will use sample statistics to estimate or make inference about parameters.
Example 1: Compute the mean of the list of numbers: 1, 5, 7, 10, 12
Answer:
Example 2: In the Spring 2012 Elementary Statistics, 161 students submitted a valid numeric value, denoted , for the number of texts in the month prior to the date this data was collected and
The average number of texts in the month prior for the Spring 2012 Elementary Statistics class at ACPHS was
_____________________________
Example 3: The MHEALTH.xlsx contains data for 40 male patients. Denote the variable for BMI (body mass index) as . Suppose...

...Mean, Mode and Median
Ungrouped and Grouped Data
Ungrouped Data refers to raw data
that has been ‘processed’; so as to
determine frequencies. The data,
along with the frequencies, are
presented individually.
Grouped Data refers to values that
have been analysed and arranged into
groups called ‘class’. The classes are
based on intervals – the range of
values – being used.
It is from these classes, are upper and
lower class boundaries found.MeanMean
The
‘Mean’ is the total of all the values in the set of data divided by the
total number of values in a set of data.
The arithmetic mean (or simply "mean") of a sample is the sum the
sampled values divided by the number of items in the sample.
x is the value of a member of the set of data
f is the frequency or number of members of the set of data
Mean=
Therefore: = 6.56
Grades
Frequency (f)
Total Value (x)
1
5
5
2
2
4
3
7
21
4
4
16
5
4
20
6
1
6
7
8
56
8
3
24
9
5
45
10
4
40
11
4
44
12
5
60
TOTALS
52
341
Mean in relation to Grouped Data
Mean in relation to grouped data
emphasizes the usage of class
intervals. Rather than the data being
presented individually, they are
presented in groupings (called
class). It is from there a midpoint is
Grade
Intervals
Frequency (f)
1-3
14
4-6
9...

...Mean, Mode and Median
Ungrouped and Grouped Data
Ungrouped Data refers to raw data
that has been ‘processed’; so as to
determine frequencies. The data,
along with the frequencies, are
presented individually.
Grouped Data refers to values that
have been analysed and arranged into
groups called ‘class’. The classes are
based on intervals – the range of
values – being used.
It is from these classes, are upper and
lower class boundaries found.MeanMean
The
‘Mean’ is the total of all the values in the set of data divided by the
total number of values in a set of data.
The arithmetic mean (or simply "mean") of a sample is the sum the
sampled values divided by the number of items in the sample.
x is the value of a member of the set of data
f is the frequency or number of members of the set of data
Mean=
Therefore: = 6.56
Grades
Frequency (f)
Total Value (x)
1
5
5
2
2
4
3
7
21
4
4
16
5
4
20
6
1
6
7
8
56
8
3
24
9
5
45
10
4
40
11
4
44
12
5
60
TOTALS
52
341
Mean in relation to Grouped Data
Mean in relation to grouped data
emphasizes the usage of class
intervals. Rather than the data being
presented individually, they are
presented in groupings (called
class). It is from there a midpoint is
Grade
Intervals
Frequency (f)
1-3
14
4-6
9...

...Mean, Median, Mode, and Range
Mean, median, and mode are three kinds of "averages". There are many "averages" in statistics, but these are, I think, the three most common, and are certainly the three you are most likely to encounter in your pre-statistics courses, if the topic comes up at all.
The "mean" is the "average" you are used to, where you add up all the numbers and then divide by thenumber of numbers. The "median" is the "middle" value in the list of numbers. To find the median, your numbers have to be listed in numerical order, so you may have to rewrite your list first. The "mode" is the value that occurs most often. If no number is repeated, then there is no mode for the list.
The "range" is just the difference between the largest and smallest values.
Find the mean, median, mode, and range for the following list of values:
13, 18, 13, 14, 13, 16, 14, 21, 13
The mean is the usual average, so:
(13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13) ÷ 9 = 15
Note that the mean is not a value from the original list. This is a common result. You should not assume that your mean would be one of your original numbers.
The median is the middle value, so I will have to rewrite the list in...

...Mean, median, and mode are differing values that furnish information regarding a set of observations. The mean is used when one desires to determine the average value for data ranked in intervals. The median is used to learn the middle of graded information, and the mode is used to summarize non-numeric data.
The mean is equal to the amount of all the data in a set divided by the number of values in that set. It is typically used with continuous figures. The result will probably not be one of the values in the data set, but is a representation of all those values. In other words, if I want to find the mean salary at a particular company, I would add together all the salaries and divide by the total number of salaries added: $50,000 + $56,000 + $54,500 = $53,500.
The problem with mean figures is they are easily slanted by one figure that stands far above or below the others. In the previous example, if I have three annual salaries of $50,000, $56,000, and $54,500, and then the company president’s salary of $260,000, I will derive an average salary of $105,125. This mean is double the actual salaries of the lower paid workers. In this case it would be more appropriate to find the median salary. To find the median salary in the previous example, we arrange the data according to value: $50,000, $54,500,...

...1.Mean and median are used as the primary measurement. Mode is seen in the first table and table 3
Appropriate measure of central tendency? Absolutely, the mean is clearly stated and many variations are introduced. Comparisons between years are used to show increases or decreases within the infant mortality rate.
2. How were measures of variation used in the study? Amongst the data collected several variations were introduced. These variations within cultures including Whites, Blacks, American Indians and so on. These variations allow us to view which, if any particular culture is experiencing a higher infant mortality rate. Which, in this case shows that blacks and non Hispanic blacks were experiencing a much higher rate of infant deaths in 2001 and 2002 respectively.
A second variation introduced on table 2 shows that multiple births are more likely to experience an infant death in contrast to a single child birth. In table 3 cause of death is used to make the determination. It appears that it is dependent on birth weight above or below a certain marker and whether “general” cause of death or congenital defects. Chromosomal abnormalities and deformations were a cause of death.
We can also conclude from the variations that are given that all races with the exception of Asians are above the standard mean. Also that abnormalities are a large portion of the total infant deaths compared to “normal”....

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