# 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. ${\sum }^{\text{​}}fm$
$53.25\left(1\right)+59.5\left(0\right)+65.5\left(4\right)+71.5\left(4\right)+77.5\left(2\right)+83.5\left(3\right)+89.5\left(4\right)+95.5\left(1\right)=1460.25$
• $\mu =\frac{\sum fm}{\sum f}=\frac{1460.25}{19}=76.86$

you are working for a bank.The bank manager wants to know the mean waiting time for all customers who visit this bank. she has asked you to estimate this mean by taking a sample . Briefly explain how you will conduct this study. assume the data set on waiting times for 10 customers who visit a bank. Then estimate the population mean. choose your own confidence level.
what marriage for 10 years
fit a least square model of y on x ? what is the regression coefficient ? x : 2 3 6 8 9 10 y : 5 6 7 10 8 11
how can we find the expectation of any function of X?
Jennifer
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Halim
what is probability
simply probability means possibility.. definition:Probability is a measure of the likelihood of an event to occur.
laraib
fit a least square model of y on x ? what is the regression coefficient ? x : 2 3 6 8 9 10 y : 5 6 7 10 8 11
Nayab
classification of data by attributes is called
qualitative classification
talal
tell me details about measure of Dispersion
Halim
Following data provided Class Frequency less than 10 10-20 5 15 10-30 25 12 40 and above Which measure of central tendency would you compute and why?
a box contains a few red and a few blue balls.one ball is drawn randomly find the probability of getting a red ball if we know that there are 30 red and 40 blue balls in the box
3/7
RICH
Total=30+40=70 P(red balls) =30/70 Therefore the answer is 3/7
Anuforo
define transport statistical unit
describe each transport statistical unit
Dennis
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identify various transport statistical units with their example
Dennis
I didn't understand about Chi- square.
explain the concept of data analysis and data processing
mean=43+37+35+30+41+23+33+31+16/10 =310/10 =31
Anuforo
43+37+35+30+41+23+33+31+16 divided by 10 =310/10 =31
Anuforo
=310/10 =31
Anuforo
In a recent survey of nurses in Region II, it was found out that the average monthly net income of nurses is Php 8,048.25. Suppose a researcher wants to test this figure by taking a random sample of 158 nurses in Region II to determine whether the monthly income has changed. Supposed further that th
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Suppose from the 1,000 freshmen that took the entrance examination, it was found out that the standard deviation was 10 a. How many students passed the test if the passing score is set 75? b. What scores comprise the middle 95% of all scores?
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