# 2.5 Measures of the center of the data  (Page 3/11)

 Page 3 / 11

## Try it

Five credit scores are 680, 680, 700, 720, 720. The data set is bimodal because the scores 680 and 720 each occur twice. Consider the annual earnings of workers at a factory. The mode is $25,000 and occurs 150 times out of 301. The median is$50,000 and the mean is $47,500. What would be the best measure of the “center”? Because$25,000 occurs nearly half the time, the mode would be the best measure of the center because the median and mean don’t represent what most people make at the factory.

## The law of large numbers and the mean

The Law of Large Numbers says that if you take samples of larger and larger size from any population, then the mean $\overline{x}$ of the sample is very likely to get closer and closer to µ . This is discussed in more detail later in the text.

## Sampling distributions and statistic of a sampling distribution

You can think of a sampling distribution as a relative frequency distribution with a great many samples. (See Sampling and Data for a review of relative frequency). Suppose thirty randomly selected students were asked the number of movies they watched the previous week. The results are in the relative frequency table shown below.

# of movies Relative Frequency
0 $\frac{5}{30}$
1 $\frac{15}{30}$
2 $\frac{6}{30}$
3 $\frac{3}{30}$
4 $\frac{1}{30}$

If you let the number of samples get very large (say, 300 million or more), the relative frequency table becomes a relative frequency distribution .

A statistic is a number calculated from a sample. Statistic examples include the mean, the median and the mode as well as others. The sample mean $\overline{x}$ is an example of a statistic which estimates the population mean μ .

## Calculating the mean of grouped frequency tables

When only grouped data is available, you do not know the individual data values (we only know intervals and interval frequencies); therefore, you cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table we can apply the basic definition of mean: mean = We simply need to modify the definition to fit within the restrictions of a frequency table.

Since we do not know the individual data values we can instead find the midpoint of each interval. The midpoint is . We can now modify the mean definition to be where f = the frequency of the interval and m = the midpoint of the interval.

A frequency table displaying professor Blount’s last statistic test is shown. Find the best estimate of the class mean.

50–56.5 1
56.5–62.5 0
62.5–68.5 4
68.5–74.5 4
74.5–80.5 2
80.5–86.5 3
86.5–92.5 4
92.5–98.5 1
• Find the midpoints for all intervals
50–56.5 53.25
56.5–62.5 59.5
62.5–68.5 65.5
68.5–74.5 71.5
74.5–80.5 77.5
80.5–86.5 83.5
86.5–92.5 89.5
92.5–98.5 95.5
• Calculate the sum of the product of each interval frequency and midpoint. ${\sum }^{\text{​}}fm$
$53.25\left(1\right)+59.5\left(0\right)+65.5\left(4\right)+71.5\left(4\right)+77.5\left(2\right)+83.5\left(3\right)+89.5\left(4\right)+95.5\left(1\right)=1460.25$
• $\mu =\frac{\sum fm}{\sum f}=\frac{1460.25}{19}=76.86$

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