Data Analysis and Statistics

When you have to interpret information, follow these steps:

• Understand the title of the graph

• Read the labels

• Analyze pictures

• Recognize scales

• Look for trends

• Use only the information on the graph, don’t use personal knowledge or opinions

Measure of Central Tendency

• Median

• Mode

• Mean

Mean

• Average of the data values• Influenced by outliers• Mean is equal to

the sum or the data values

the number of data values in the data set

Median

• Middle value of a data set

• Average of the two middle values

• Not influenced by outliers

• Based on relative size of data set, not on the actual values

Mode

• Value that occurs most frequently

• Can be one, more than one, or no mode

• Only appropriate measure of central tendency for data that is strictly nonnumeric

• Based on relative frequency rather than all the values in the set

Richard has participated in eight track meets so far this season. His running times for the 440-meter race have been 73, 63, 68, 64, 69, 61, 66, and 64 seconds. What is Richard’s median running time for the eight meets?

A. 64 seconds

B. 65 seconds

C. 66 seconds

D. 66.5 seconds

A. 64 seconds

B. 65 seconds

C. 66 seconds

D. 66.5 seconds

First put the numbers in order.61, 63, 64, 64, 66, 68, 69, 73Since there is no middle, average 64 and 66.

Measure of Dispersion

• Range

• Standard deviation

• Variance

Range

• Difference between maximum value and minimum value

• Should have the same units as those of the data values from the data set

Standard Deviation

• A measure of the dispersion of a set of data from its mean. The more spread apart the data is, the higher the deviation.

• Standard deviation can also be calculated as the square root of the variance.

x

Variance

• Square of the standard deviation of the population.

Regression

r

• Correlation coefficient– The closer is to 1, the more perfect is the linear

relationship between x and y.– If r is close to zero, there is little or no linear

relationship.

r

Normal Distributions

68-95-99.7 Rule•68% of the values are within 1 standard deviation of the mean•95% of the values are within 2 standard deviations of the mean•99.7% of the values are within 3 standard deviations of the mean

A. 2.5%

B. 16%

C. 34%

D. 68%

The lifetime of a certain type of disposable razor is normally distributed with a mean of 16.8 shavings and a standard deviation of 2.4 shavings. What percentage of disposable razors of this type will last more than 19.2 shavings?

A. 2.5%

B. 16%

C. 34%

D. 68%

First you need to find the z-score for 19.2. The z-score is 1. Therefore, 19.2 is 1 standard deviation above the mean. Now find the percentage of the normal distribution that is 1 standard deviation above the mean. If we were to look at the normal curve we would see that from 16.8 – 2.4 to 16.8 + 2.4 there is 68% of the data. Which means the remaining 32% of the graph contains the rest of the data. However, because the graph is symmetric, half of the 32% is below 16.8 – 2.4. Therefore, 16% is above 16.8 + 2.4.

Z-Score

• Number of standard deviations away from the mean

( )x valuez score

Other Key Words

• Quartiles ~ four portions

• Skewness ~ lopsidedness– Positively skewed (longer tail to the right)– Negatively skewed (longer tail to the left)

Donna scored at the 75th percentile on a multiple-choice history exam. The best interpretation of this information is that

A. Donna answered 75% of the questions on the test correctly

B. Only 25% of the other students did worse on the test than did Donna

C. Donna answered 75 questions correctly

D. Donna did as well as or better than 75% of the students who took the exam

A. Donna answered 75% of the questions on the test correctly

B. Only 25% of the other students did worse on the test than did Donna

C. Donna answered 75 questions correctly

D. Donna did as well as or better than 75% of the students who took the exam

The 75th percentile is a value at or below which 75% of the data fall. Therefore, the best interpretation of Donna’s score is that she did as well as or better than 75% of the students who took the exam.