Stats: Ch 2

When a data set has two different modes, it is known as what type of distribution?

A bimodal distribution is where the data set has two different modes.

What do we call areas in a data set where no observations have been made?

A gap is an area in the data set where no observations have been made.

What is a graphical representation of the distribution of data?

A histogram is a graphical representation of the distribution of data

Median

as the middle score after the scores have been arranged in numerical order

Mode

as the most often occurring value

With what kind of value set does the mean provide the best representation?

data sets with numbers that are close together

a type of statistics used to describe and summarize values in a data set.

descriptive statistics

A group of students decide to hold a staring contest. The student that can stare without blinking for the longest amount of time is the winner. Using the standard competition ranking, who scored 5th place? Jon - 32 sec tucker - 12 sec travis - 44 sec sydney - 28 sec isabel - 24 sec beverly - 8 sec lisa - 28 sec

in this case, we need to organize the data from the longest time to the shortest time because the student that can stare for the longest time is the winner. Taking numbers only, we will organize them from greatest to least: 44, 32, 28, 28, 24, 12, 8 Now, to find 5th place using standard competition rating, we look at the times. Note that there are two times that are the same. The 2 students who scored 28 seconds both receive 3rd place. Under SCR we skip the next rank (4th) so that the student that scored 24 seconds is in 5th place. Thus, according to the table, Isabel is in 5th place

Outlier

is a value that is much larger or smaller than the other values in a data set, or a value that lies outside the given data set

Median

is the midpoint value of a data set, where the values are arranged in ascending or descending order. The median can be used to find the center of data when the numbers in the data set contain one or more outliers.

Mean

is the sum of the numbers in a data set divided by the total number of values in the data set.

Phyllis is also in Jonathan's psychology class. She knows she scored in the 90th percentile on his psychology test. The professor gives the students the following list of grades (without names): 51, 37, 87, 95, 99, 78, 63, 96, 68, 84, 92 Assuming that there are 11 students in the class, what was Phyllis's test score?

11*.9 = 9.9 Order the number line and round up to ten and her score is the tenth one in or 96

The following data set represents the average annual temperatures for 10 European countries in 2016 (Celsius). Calculate the range. 1.75, 9.8, 7.55, 10.55, 5.8, 9.45, 19.2, 18.45, 16.2, 5.8

17.45 - Because the range is the difference between the highest and lowest values in a data set, the range of this set is the difference of {19.2-1.75}, which is equal to 17.45.

Find the second quartile, or the median, for the following data set: 7, 9, 13, 4, 18, 3, 9, 10, 15, 8, 2, 6, 9

2, 3, 4, 6, 7, 8, 9, 9, 9, 10, 13, 15, 18 The median of the data set is the first 9

Find the first quartile for the following data set: 7, 9, 13, 4, 18, 3, 9, 10, 15, 8, 2, 6, 9

2, 3, 4, 6, 7, 8, 9, 9, 9, 10, 13, 15, 18 The median of the data set is the first 9. The first half of the data includes 2, 3, 4, 6, 7, 8 (everything below the median). The median of the first half is 5 [(4 + 6)/2].

Find the interquartile range for the following data set: 7, 9, 13, 4, 18, 3, 9, 10, 15, 8, 2, 6, 9

2, 3, 4, 6, 7, 8, 9, 9, 9, 10, 13, 15, 18 The median of the data set is the first 9. The first half of the data includes 2, 3, 4, 6, 7, 8 (everything below the median). The median of the first half or first quartile is 5 [(4 + 6)/2]. The second half of the data includes 9, 9, 10, 13, 15, 18. The median of the second half of the data or third quartile is 11.5 [(10 + 13)/2] The interquartile range is the difference between the first and third quartiles: 11.5 - 5 = 6.5.

During the first four Summer Olympic games attended by the United States, these medal counts were awarded. Calculate the range of medals won during these Olympic Games. [20,47,239,47] {highest number minus lowest number}

219

During the first four Summer Olympic games attended by the United States, these medal counts were awarded. If there is an outlier, identify it from the number of medals won during these Olympic Games. [20,47,239,47] {is the highest #}

239

Find the maximum in the data set: 27, 26, 33, 45, 38, 38, 58, 54, 19, 22, 31

58

During the first four Summer Olympic games attended by the United States, these medal counts were awarded. Calculate the median number of medals won during these Olympic Games. [20,47,47,239] {add the two middle #s/2 after arranging the #s}

47

During the first four Summer Olympic games attended by the United States, these medal counts were awarded. Calculate the mode number of medals won during these Olympic Games. [20,47,239,47]{repeated #}

47

What is the minimum in the data set? 105, 88, 93, 102, 97, 93, 96, 86, 88, 82, 79, 85, 77, 94, 91, 87

77

The following data set represents the minimum wage for 1990-2000. Calculate the median. (As this is money, round to two decimal places.) 8.25, 9.80, 7.25, 7.25, 8.44, 8.75, 8.44, 9.50, 9.60, 9.50

8.60 - Because this data set has ten numbers, there is no single median. The median is found by calculating the average of the two middle terms, 8.44 and 8.75

During the first four Summer Olympic games attended by the United States, these medal counts were awarded. Calculate the mean number of medals won during these Olympic Games. (Round it to the nearest whole number.) [20,47,239,47] {add the #'s /4}

88

A data set contains the following numbers: 9, 1, 10, 12, 13, 15, 12, 8 Which of the following options would describe the data set best?

Because median is best used for a data set with numbers that have a few larger or smaller numbers, as well as several numbers close together, the above data set would best be described as the median.

Jonathan knows he scored in the 25th percentile on his psychology test. The professor gives the students the following list of grades (without names): 51, 37, 87, 95, 99, 78, 63, 96, 68, 84, 92 Assuming that there are 11 students in the class, what was Jonathan's test score?

Begin by multiplying the number of scores by the desired percentile. 11*.25 = 2.75. We put the numbers in order and round up to get a whole number then we know that his score is the third in the line, or 63.

Which of the following terms best describes the distribution in the chart below? EX2

Bimodal - The chart shows a bimodal distribution because the data set has two different modes. Remember that the mode of a data set is the value that appears the most frequently in the data set. In the above chart we see two different modes, or peaks, at almost 16.

Find the sample variance of the following data set: 2, 3, 4, 7, 10, 1, 2

By following the steps, Find the mean of the set of data. 4.14 Subtract each number from the mean. Square the result. Add the results together. Then divide by the total number of numbers in the data set, minus one. 6 You should get 10.48

Find the population variance of the following data set: 11, 12, 55, 4, 17, 13, 19

By following the steps, Find the mean of the set of data. Subtract each number from the mean. Square the result. Add the results together. Divide the result by the number of number in the data set, you should get approximately 239.

What is a single number that summarizes an entire data set?

Center of data

What is the difference between the upper quartile value and the lower quartile value?

Interquartile range

To find the median of a set of values, what is the first thing one must do?

arrange the values in ascending or descending order

What is a system of ordering in which the mathematical values that are equal are given the mean of the ranking positions?

Fractional ranking is a system of ordering in which the mathematical values that are equal are given the mean of the ranking positions.

In the linear transformation formula, what does x represent?

In the linear transformation formula, x = the number in the data set.

If a data set has two different modes, then what is it?

Data is bimodal when the data set has two different modes.

Find the variance for the following data set: 101, 106, 125, 142, 78, and 109

First find the mean: (101 + 106 + 125 + 142 + 78 + 109) / 6 = 110 Now subtract the mean from each value: 101 - 110 = -9 106 - 110 = -4 125 - 110 = 15 142 - 110 = 32 78 - 110 = -32 109 - 110 = -1 Square each of these values and find the mean of the sum of these numbers: (81 + 16 + 225 + 1024 + 1024 + 1) / 6 = 395

Find the standard deviation for the following data set: 101, 106, 125, 142, 78, and 109

First find the mean: (101 + 106 + 125 + 142 + 78 + 109) / 6 = 110 Now subtract the mean from each value: 101 - 110 = -9 106 - 110 = -4 125 - 110 = 15 142 - 110 = 32 78 - 110 = -32 109 - 110 = -1 Square each of these values and find the mean of the sum of these numbers: (81 + 16 + 225 + 1024 + 1024 + 1) / 6 = 395 To find the standard deviation take the square root of the variance: √395 = 19.9

Find the variance for the following data set: (rounding the answer to the nearest tenth). 16, 20, 14, 24, 16, 19, 30, and 8

Following the steps, find the variance by subtracting the mean (18.375) from each number, then squaring each of those numbers (rounding to the thousandths). Once that is finished, determine the average by adding them together and dividing by the total number of numbers. For this number set the answer is rounded to the answer 38.5

Find the range for one standard deviation in the following data set: 16, 20, 14, 24, 16, 19, 30, and 8

Following the steps, find the variance by subtracting the mean (18.375) from each number, then squaring each of those numbers. Once that is finished, determine the average by adding them together and dividing by the total number of numbers. The variance is 38.35. To get the standard deviation find the square root of the variance. (SD = 6.2) To calculate this range, you will add and subtract the standard deviation to the mean. 18.375 ± 6.2. For this number set, you will get 12.175 to 24.575

Find the standard deviation for the following data set: 16, 20, 14, 24, 16, 19, 30, and 8

Following the steps, find the variance by subtracting the mean from each number, then squaring each of those numbers. Once that is finished, determine the average by adding them together and dividing by the total number of numbers. To get the standard deviation take that average and find the square root. For this number set the answer is 6.2

When a variable is multiplied by a constant and then added to a constant, this is known as what?

Linear transformation is when a variable is multiplied by a constant and then added to a constant. When using linear transformations on a data set, all variables in the data set are transformed.

What is the largest mathematical value in a data set?

Maximum

What is the sum of the numbers in a data set divided by the total number of values in the data set?

Mean

a description of the variability or how spread out the data is in a data set.

Measure of variation

For the following data set, select the best method for summarizing the data. 3, 3, 4, 5, 2, 9, 5, 2, 3, 4

Median

What is the midpoint value of a data set, where the values are arranged in ascending or descending order?

Median

What is the smallest mathematical value in a data set?

Minimum

Sample

as a section or part of the population and would be anywhere from 1% to 99% of the city population.

What is a system of ordering where each mathematical value is given a certain position in a sequence of numbers, where no positions are equal?

Ordinal ranking is a system of ordering where each mathematical value is given a certain position in a sequence of numbers where no positions are equal.

What do we call a value that is much smaller or larger than the other values in a data set?

Outlier

What is a value that is much larger or smaller than the other values in a data set, or a value that lies outside the given data set?

Outlier

What is a measure that indicates what percent of the given population scored at or below the measure?

Percentile

Which of the following describes the best approach to finding the median of an even set of numbers?

Place numbers in ascending or descending order, calculate the average of the middle terms

All the members of a specified group are known as:

Population

What is a group of values and/or means that divide a data set into quarters, or groups of four?

Quartile

Population

as the complete collection to be studied

What is the relationship between two mathematical values, where each value can be less than, greater than, or equal to the second value?

Ranking is the relationship between two mathematical values where each value can be less than, greater than, or equal to the second value.

A part of a population used to describe the whole group is known as:

Sample

A graph that peaks to the left or the right of the center is known as _____.

Skewed - If your visual representation of a data set is not symmetrical, then it might be skewed, which is where the shape of a graph peaks to the left or the right of the center.

What is a ranking system in which the mathematical values that are equal are given equal rank and the next, lesser value is given the next highest rank?

Standard competition ranking

A statistician makes a graph, which includes a shape that's almost mirrored perfectly across a line, but not quite. How should we describe the shape?

Symmetrical - The representation will be symmetrical when the shape created is mirrored nearly perfectly across a line

The mean of an original data set is 5. After linear transformation, the mean of the data set is now 20. Which of the following equations is a possible representation of the linear transformation?

The mean of an original data set is 5. After linear transformation, the mean of the data set is now 20. Which of the following equations is a possible representation of the linear transformation?

What is the variance of the data set below? 3, 7, 9, 11, 11, 11, 15, 17, 18, 24

The mean of the group is 12.6. Subtract each number from 12.6 and square it: 9.6 92.16 5.6 31.36 3.6 12.96 1.6 2.56 1.6 2.56 1.6 2.56 -2.4 5.76 -4.4 19.36 -5.4 29.16 -12.4 129.96 Now add them together to get 328.40. Divide by 10 = 32.84

Measures of central tendency

are a set of descriptive measures that indicate the typical score. When conducting research, we need or want to know what is most likely going to happen.

is frequently used with data sets that include categorical data, or data that can be organized into groups, but does not have mathematical meaning.

The mode

The value that occurs the most frequently in a data set is known as the _____.

The mode of a set of numbers is the terms which occurs most often in that data set. For example, in the set {4, 5, 6, 6}, the mode would be 6.

What is the range of the data set below? 3, 7, 9, 11, 11, 11, 15, 17, 18, 24

The range is the difference between the highest and lowest values in a data set. The numbers are arranged from least to greatest: 3, 7, 9, 11, 11, 11, 15, 17, 18, 24 Now take the lowest number and the highest number and find the difference: 24-3= 21.

What is the measure of how far the numbers in a data set are away from the mean or the median?

The spread in data is the measure of how far the numbers in a data set are away from the mean or median.

The following data set represents the average number of children per family for 10 countries in 2014. Calculate the mean number of children per family. 5.8, 6.1, 1.9, 1.4, 2.6, 2.8, 1.3, 4.4, 4.4, 1.7

The sum of the 10 terms in the data set is 32.4. This number divided by 10, which is the number of individual terms, the average number of children per family in 2014 among these countries was 3.24.

What are three methods you can use to find the spread in data?

There are three methods one can use to find the spread in the data: range, interquartile range, and variance.

Mean

as the data's average score of the sample

Which of the following terms best describes the distribution in the chart below? EX1

Unimodal - There is only one mode, or peak, in the chart above. The mode is almost at 16

When a data set has a single mode, it is known to have what type of distribution?

Unimodal - Unimodal distribution is when the data set has a single mode

All of the students in Jasmine's class have had their height measured. Jasmine is the 12th tallest person out of 560 students. What percentile is Jasmine's height in?

Using our formula (k + .5r) / n = p we know that k=548 (number of students shorter than her) r=1 (assuming that no other student shares her height) and n=560 (the number of total students) plug our numbers into the equation (548 + .5(1)) /560 = p solve and round up for 98th

How far a set of numbers are spread out is known as:

Variance

The equation for linear transformation is:

We can transform the data in a data set by using the following formula for linear transformations: a + bx.

When a data set is visually symmetrical, what occurs all at the same point?

When a data set is symmetrical, then the mean, median and mode all occur in the same point.

If a data set undergoes a linear transformation with positive values of a and b, why might both the mean and the standard deviation of the data set increase?

When both the mean and standard deviation increases, the value of each data point increases. This mean that our data set is now spread farther out.

Select the word or phrase that best fills in the blank: If the index is not a _____, round the number up, then count the values in the data set from least to greatest until you reach the index.

Whole number

Center of data

a single number that summarizes the entire data set. You can find the center of data using either the mean or the median of the data set.

The following data represents the test scores of eight students in Mr. Miller's science class. 92%, 91%, 89%, 95%, 45%, 88%, 90%, 91% The student who got a 45% is _____.

an outlier

All of the following statements are correct, EXCEPT:

an outlier is likely to skew the median more than the mean of a sample

If there are two modes, you should _____.

report both

is when the tail of a distribution on a graph is longer than the other. If the tail is longer towards the negative side of a number line, it is left-skewed. If the tail is longer on the positive side of a number line, it is right-skewed. The more a distribution is skewed, either right or left, the less accurate the mean

skewed distribution

or bell-curve, the mean, median, and mode are all at the center of the distribution

symmetrical distribution

The best predictor of a score is the _____.

the mean

You have a set of numbers you are working with to provide some data. You add them all up and divide that sum by the total number of values in your set. What statistic did you just produce?

the mean

Which of the following is least likely to be reported?

the mode

is found by subtracting the highest and lowest value.

the range

Beth is an apartment complex manager, she wants to plan some activities for the residents that will increase a sense of community among them. She knows that age appropriate events will attract more residents. According to the rental applications, the ages of her residents are as follows: 60, 48, 26, 20, 66, 49, 22, 25, 63, 59 From this, Beth knows the mean age of her residents is 43.8. She will schedule activities that individuals in their mid-forties (43-46) will enjoy. Would finding the median of this data set be more helpful for Beth? Why?

yes; the ages of her residents are skewed from the mean, around half of the residents are much younger and around half are older than 43.8