Free Essay

Exploring Averages

In: Science

Submitted By lucknicole
Words 1569
Pages 7
Quantitative Literacy
Mat143 LN12
Group Black
Nicole Luck
Lisa Becquet
Sandra Soto
Heather Wicker
Jared Gunter

Exploring the Average I. Background Information

* The three most common measures of center:

Mean
The mean is the best-known and most widely used measure of central tendency. It is what most people call the "average." The mean is calculated by adding up all the scores and dividing the result by the number of scores in the distribution.

Median
The median represents the exact middle of a distribution so that half of the cases are above it and half are below it. It is the middle case in a distribution when the scores are arranged in order from lowest to highest.

Mode
The mode is the category or score with the highest frequency in the distribution. In other words, it is the most common score, or the score that appears the highest number of times in a distribution.

* Skewness
A distribution is said to be skewed when the data points cluster toward one side of the scale than the other, creating a curve that is not symmetrical. In other words, the right and the left side of the distribution are shaped differently from each other. There are two types of skewed distributions. A distribution is positively skewed if the scores fall toward the lower side of the scale and there are very few higher scores. Positively skewed data is also referred to as 'skewed to the right' because that is the direction of the 'long tail end' of the chart. A distribution is negatively skewed if the scores fall toward the higher side of the scale and there are very few low scores.

* Mean, median and mode of a frequency distribution:

http://www.youtube.com/watch?v=vqHEFqKU5gQ

* Creating a bar (column) chart using excel:

http://www.youtube.com/watch?v=Ljc9aizzjsM

II. Exploration

A. Imagine that you were interested in studying the annual income of students one year after they have completed their Master of Business Administration (MBA). You collect data from 400 graduates and find that their yearly income ranges from $20,000 to $150,000. The table below summarizes the data that you have collected. Income (thousands) | Frequency | less than 31 | 3 | 31-40 | 50 | 41-50 | 70 | 51-60 | 87 | 61-70 | 70 | 71-80 | 40 | 81-90 | 30 | 91-100 | 20 | 101-110 | 11 | 111-120 | 9 | 121-130 | 5 | 131-140 | 3 | 141-150 | 2 |

1. Construct a bar (column) chart in excel. Insert the graph here.

2. Describe the shape of the distribution.
The distribution is skewed to the right.

3+50+70+87+70+40+30+20+11+9+5+3+2

3. Calculate the mean.
3+50+70+87+70+40+30+20+11+9+5+3+2= 400/ 13 = 30.8 or 31

4. Estimate the median.
3, 50, 70, 87, 70, 40, 30, 20, 11, 9,5 3, 2 The median is 30

5. Estimate the mode.
The mode is 3 and 70

6. Is the mean greater than, less than, or equal to the median? Explain why this is true.
The mean is just slightly more than the median but only by a little bit. I think this is because the date was skewed towards the right, which is what typically tends to happen.

7. What is the value of the mode in relation to the mean and median?
It represents the group that had the lowest frequency. This would be correct because bar graphs that tend to be skewed to the right, the mode tends to be less than the mean and the median.

8. What can you conclude about the annual income of students one year after they have completed their MBA?
That most tend to earn their income in the 31k to 50k range after they have completed their MBA, earning them a better living and more positive aspect on life.

B. Let's say you are also interested in examining the number of applications each graduate completed before they found their current job. Using data collected from the same 400 graduates, you find that the number of applications they completed ranges from 1 to 15. The table below summarizes the data that you have collected. Applications Completed | Frequency | 1 | 1 | 2 | 2 | 3 | 2 | 4 | 3 | 5 | 5 | 6 | 8 | 7 | 10 | 8 | 25 | 9 | 40 | 10 | 60 | 11 | 80 | 12 | 70 | 13 | 50 | 14 | 33 | 15 | 11 | | | 1. Construct a bar chart in excel. Insert the graph here.

Series 4 Indicates Frequency(gold) and Series 5 Indicates Applications(blue) completed.

2. Describe the shape of the distribution.
The distribution has one peak ad skewed left.

3. Calculate the mean. 1x1+2x2+3x2+4x3+5x5+6x8+7x10+8x25+9x40+10x60+11x80 +12x70 +13x50 +14x33+15x11
1+2+2+3+5+8+10+25+40+60+80+70+50+33+11 =
4,583
400
=11.46

4. Calculate the median.
1+2+2+3+5+8+10+25+40+60+80+70+50+33+11= 400 (n/2) + 1= (400/2) + 1= 200 + 1= 201 is the median

5. Is the mean greater than, less than, or equal to the median? Explain why this is true.
The mean is less than the median and this is common in charts skewed to the left, or also known as a negative skew.

6. What is the value of the mode in relation to the mean?
The value that is most common in a row of data is the mode. In other words, it is the value with the highest frequency. The modal group in this table is group 11, which had the highest frequency of 80. The mean is also 11.47, making it closely related to the mode.

7. What can you conclude about the number of applications each graduate completed before they found their current job.
I can conclude that while putting in more applications increases your chances of finding a job, it is also situational and all depends on your skills and background.

C. Finally, you would like to study GRE scores of the MBA graduates. GRE scores | Frequency | below 960 | 31 | 960-1059 | 48 | 1060-1159 | 64 | 1160-1259 | 98 | 1260-1359 | 72 | 1360-1459 | 64 | above 1459 | 23 |

1. Construct a bar chart in excel. Insert the graph here.

2. Describe the shape of the distribution.
The shape of distribution is unimodal, having only one peak/mode. It is also right-skewed.

3. Calculate the mean.
1209.167

4. Estimate the median.
1209

5. Estimate the mode.
1359

6. Is the mean greater than, less than, or equal to the median? Explain why this is true.
The mean is greater than the median, because if you add all values and then divide by the amount of values you get the mean, which is 1209.167. The median, however, is the middle value which we find by adding 1159 +1259 = 2418, then divide by two = 1209

7. What can you conclude about GRE scores of the MBA graduates?
Most MBA graduates had scores that were in the middle; less MBA graduates scored low or very high.

III. Extensions
1. Give an example of a distribution that is naturally normally distributed. Explain.
Heights of people and blood pressure measurements. Mean=median=mode, symmetry is about in the center and 50% of values less than the mean and 50% greater than the mean. There is a narrow range.

2. Give an example of a distribution that is naturally skewed left. Explain. The age people die from heart disease. There are far more deaths at older ages than younger so the data is skewed left, with few deaths on left side and more on the right.

3. Give an example of a distribution that is naturally skewed right. Explain.
Number of books read by a 5th grade in a school year. Most will read a moderate amount of books but few will read far more than others. These students will therefore be outliners with high values for the number of books read creating a tail on the right side of distribution.

4. Give an example of a data set where the mode is the only possible measure of center.
Termperature in a given week.

5. What is the best measure of center? Explain .
It really just depends on what you are finding the average of. Mean is good when the numbers have a low variable of change. Median works well when the variable is larger. Mode works well when you want to know what is the most coomon.

6. Suppose there are 7 people who graduate from some university with degrees in communications. They all get jobs, and their salaries are: $27,000 $29,000 $33,000 $34,000 $35,000 $39,000 $5,000,000
What is their “average” salary? Which measure of center did you use? Explain why.

34,000. I used median because the 5,000,000 salary would make the average very high which would be wrong because the majority if these workers make considerably lower than 5,000,00.

What does this saying mean?
“A statistician drowned while crossing a river that was on average six inches deep”.
If they used mean for the average there may had been several deep areas but if the shallow areas were much more common he could have fell into a large hole/deep area of the water.

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