<< Chapter < Page Chapter >> Page >

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.

Got questions? Get instant answers now!

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 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 5 30
1 15 30
2 6 30
3 3 30
4 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 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 = d a t a   s u m n u m b e r   o f   d a t a   v a l u e s 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 l o w e r   b o u n d a r y + u p p e r   b o u n d a r y 2 . We can now modify the mean definition to be M e a n   o f   F r e q u e n c y   T a b l e = f m f 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.

Grade Interval Number of Students
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
Grade Interval Midpoint
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. f m
    53.25 ( 1 ) + 59.5 ( 0 ) + 65.5 ( 4 ) + 71.5 ( 4 ) + 77.5 ( 2 ) + 83.5 ( 3 ) + 89.5 ( 4 ) + 95.5 ( 1 ) = 1460.25
  • μ = f m f = 1460.25 19 = 76.86
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Introductory statistics' conversation and receive update notifications?

Ask