Mean, median and mode

Measures of the Location of the Data

The common measures of location are quartiles and percentiles.

Quartiles are special percentiles. The first quartile, Q₁, is the same as the 25^th percentile, and the third quartile, Q₃, is the same as the 75^th percentile. The median, M, is called both the second quartile and the 50^th percentile.

To calculate quartiles and percentiles, you must order the data from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths. Recall that a percent means one-hundredth. So, percentiles mean the data is divided into 100 sections. To score in the 90^th percentile of an exam does not mean, necessarily, that you received 90 percent on a test. It means that 90 percent of test scores are the same as or less than your score and that 10 percent of the test scores are the same as or greater than your test score.

Percentiles are useful for comparing values. For this reason, universities and colleges use percentiles extensively. One instance in which colleges and universities use percentiles is when SAT results are used to determine a minimum testing score that will be used as an acceptance factor. For example, suppose Duke accepts SAT scores at or above the 75^th percentile. That translates into a score of at least 1220.

Percentiles are mostly used with very large populations. Therefore, if you were to say that 90 percent of the test scores are less, and not the same or less, than your score, it would be acceptable because removing one particular data value is not significant.

The median is a number that measures the center of the data. You can think of the median as the middle value, but it does not actually have to be one of the observed values. It is a number that separates ordered data into halves. Half the values are the same number or smaller than the median, and half the values are the same number or larger. For example, consider the following data:
1, 11.5, 6, 7.2, 4, 8, 9, 10, 6.8, 8.3, 2, 2, 10, 1
Ordered from smallest to largest:
1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5

When a data set has an even number of data values, the median is equal to the average of the two middle values when the data are arranged in ascending order (least to greatest). When a data set has an odd number of data values, the median is equal to the middle value when the data are arranged in ascending order.

\[\]

Condensed — the full section is in OpenStax Statistics.

A Formula for Finding the

If you were to do a little research, you would find several formulas for calculating the k^th percentile. Here is one of them.

k = the k^th percentile. It may or may not be part of the data.

i = the index (ranking or position of a data value)

n = the total number of data

• Order the data from smallest to largest.
• Calculate \(i=\frac{k}{100}(n+1)\text{.}\)
• If i is an integer, then the k^th percentile is the data value in the i^th position in the ordered set of data.
• If i is not an integer, then round i up and round i down to the nearest integers. Average the two data values in these two positions in the ordered data set. The formula and calculation are easier to understand in an example.

Example

Try it.

Listed are 29 ages for Academy Award-winning best actors in order from smallest to largest:
18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77

1. Find the 70^th percentile.
2. Find the 83^rd percentile.

Solution

1. • k = 70
   • i = the index
   • n = 29
   i = \(\frac{k}{100}\) (n + 1) = (\(\frac{70}{100}\))(29 + 1) = 21. This equation tells us that i, or the position of the data value in the data set, is 21. So, we will count over to the 21^st position, which shows a data value of 64.
2. • k = 83^rd percentile
   • i = the index
   • n = 29
   i  = \(\frac{k}{100}\) (n + 1) = (\(\frac{83}{100}\))(29 + 1) = 24.9, which is not an integer. Round it down to 24 and up to 25. The age in the 24^th position is 71, and the age in the 25^th position is 72. Average 71 and 72. The 83^rd percentile is 71.5 years.

A Formula for Finding the Percentile of a Value in a Data Set

• Order the data from smallest to largest.
• x = the number of data values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile.
• y = the number of data values equal to the data value for which you want to find the percentile.
• n = the total number of data.
• Calculate \(\frac{x+.5y}{n}\)(100). Then round to the nearest integer.

Example

Try it.

Listed are 29 ages for Academy Award-winning best actors in order from smallest to largest:
18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77

1. Find the percentile for 58.
2. Find the percentile for 25.

Solution

1. Counting from the bottom of the list, there are 18 data values less than 58. There is one value of 58.

   x = 18 and y = 1.\(\frac{x+.5y}{n}\)(100) = \(\frac{18+.5(1)}{29}\)(100) = 63.80. Fifty-eight is the 64^th percentile.

2. Counting from the bottom of the list, there are three data values less than 25. There is one value of 25.

   x = 3 and y = 1.\(\frac{x+.5y}{n}\)(100) = \(\frac{3+.5(1)}{29}\)(100) = 12.07. Twenty-five is the 12^th percentile.

Interpreting Percentiles, Quartiles, and Median

A percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest. Percentages of data values are less than or equal to the pth percentile. For example, 15 percent of data values are less than or equal to the 15^th percentile.

• Low percentiles always correspond to lower data values.
• High percentiles always correspond to higher data values.

A percentile may or may not correspond to a value judgment about whether it is good or bad. The interpretation of whether a certain percentile is good or bad depends on the context of the situation to which the data apply. In some situations, a low percentile would be considered good; in other contexts a high percentile might be considered good. In many situations, there is no value judgment that applies. A high percentile on a standardized test is considered good, while a lower percentile on body mass index might be considered good. A percentile associated with a person's height doesn't carry any value judgment.

Understanding how to interpret percentiles properly is important not only when describing data, but also when calculating probabilities in later chapters of this text.

Measures of the Center of the Data

The center of a data set is also a way of describing location. The two most widely used measures of the center of the data are the mean (average) and the median. To calculate the mean weight of 50 people, add the 50 weights together and divide by 50. To find the median weight of the 50 people, order the data and find the number that splits the data into two equal parts. The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center.

When each value in the data set is not unique, the mean can be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. The letter used to represent the sample mean is an x with a bar over it (pronounced “x bar”): \(\). The sample mean is a statistic.

The Greek letter μ (pronounced "mew") represents the population mean. The population mean is a parameter. One of the requirements for the sample mean to be a good estimate of the population mean is for the sample taken to be truly random.

To see that both ways of calculating the mean are the same, consider the following sample:
1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 4

\[\overset{\bar}{x}=\frac{1+1+1+2+2+3+4+4+4+4+4}{11}=2.7\]\[\overset{\bar}{x}=\frac{3(1)+2(2)+1(3)+5(4)}{11}=2.7\text{.}\]

In the second example, the frequencies are 3(1) + 2(2) + 1(3) + 5(4).

You can quickly find the location of the median by using the expression \(\frac{n+1}{2}\).

Example

Statistics exam scores for 20 students are as follows:

50, 53, 59, 59, 63, 63, 72, 72, 72, 72, 72, 76, 78, 81, 83, 84, 84, 84, 90, 93

Try it.

Find the mode.

Solution

The most frequent score is 72, which occurs five times. Mode = 72.

Condensed — the full section is in OpenStax Statistics.

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 \(\overset{\bar}{x}\) of the sample is very likely to get closer and closer to µ. This law 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 Chapter 1: Sampling and Data for a review of relative frequency. Suppose 30 randomly selected students were asked the number of movies they watched the previous week. The results are in the relative frequency table shown below.

Number of Movies	Relative Frequency
0	\[\frac{5}{30}\]
1	\[\frac{15}{30}\]
2	\[\frac{6}{30}\]
3	\[\frac{3}{30}\]
4	\[\frac{1}{30}\]

A relative frequency distribution includes the relative frequencies of a number of samples.

Recall that 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 \(\overset{\bar}{x}\) is an example of a statistic that 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 know only 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 = \(\frac{data\ sum}{number\ of\ data\ values}\text{.}\) 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 \(\frac{lower\ boundary+upper\ boundary}{2}\). We can now modify the mean definition to be \(Mean\ of\ Frequency\ Table=\frac{\sum fm}{\sum f}\text{,}\) where f = the frequency of the interval, m = the midpoint of the interval, and sigma (∑) is read as "sigma" and means to sum up. So this formula says that we will sum the products of each midpoint and the corresponding frequency and divide by the sum of all of the frequencies.

Example

Try it.

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

Solution

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

Measures of the Center of the Data

Use the following information to answer the next three exercises: The following data show the lengths of boats moored in a marina. The data are ordered from smallest to largest: 16, 17, 19, 20, 20, 21, 23, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 29, 30, 32, 33, 33, 34, 35, 37, 39, 40

Use the following information to answer the next three exercises: Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars, 19 generally sell four cars, 12 generally sell five cars, nine generally sell six cars, and 11 generally sell seven cars. Calculate the following.

Skewness and the Mean, Median, and Mode

Consider the following data set:
4, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 10

This data set can be represented by the following histogram. Each interval has width 1, and each value is located in the middle of an interval.

The histogram displays a symmetrical distribution of data. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other. The mean, the median, and the mode are each seven for these data. In a perfectly symmetrical distribution, the mean and the median are the same. This example has one mode (unimodal), and the mode is the same as the mean and median. In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median.

The histogram for the data: 4, 5, 6, 6, 6, 7, 7, 7, 7, 8 is not symmetrical. The right-hand side seems chopped off compared to the left-hand side. A distribution of this type is called skewed to the left because it is pulled out to the left. A skewed left distribution has more high values.

The mean is 6.3, the median is 6.5, and the mode is seven. Notice that the mean is less than the median, and they are both less than the mode. The mean and the median both reflect the skewing, but the mean reflects it more so. The mean is pulled toward the tail in a skewed distribution.

The histogram for the data: 6, 7, 7, 7, 7, 8, 8, 8, 9, 10 is also not symmetrical. It is skewed to the right. A skewed right distribution has more low values.

The mean is 7.7, the median is 7.5, and the mode is seven. Of the three statistics, the mean is the largest, while the mode is the smallest. Again, the mean reflects the skewing the most.

Condensed — the full section is in OpenStax Statistics.

Skewness and the Mean, Median, and Mode

Use the following information to answer the next three exercises. State whether the data are symmetrical, skewed to the left, or skewed to the right.

Measures of the Spread of the Data

An important characteristic of any set of data is the variation in the data. In some data sets, the data values are concentrated closely near the mean; in other data sets, the data values are more widely spread out from the mean. The most common measure of variation, or spread, is the standard deviation. The standard deviation is a number that measures how far data values are from their mean.

The standard deviation

• provides a numerical measure of the overall amount of variation in a data set and
• can be used to determine whether a particular data value is close to or far from the mean.

The standard deviation is always positive or zero. The standard deviation is small when all the data are concentrated close to the mean, exhibiting little variation or spread. The standard deviation is larger when the data values are more spread out from the mean, exhibiting more variation.

Suppose that we are studying the amount of time customers wait in line at the checkout at Supermarket A and Supermarket B. The average wait time at both supermarkets is five minutes. At Supermarket A, the standard deviation for the wait time is two minutes; at Supermarket B, the standard deviation for the wait time is four minutes.

Because Supermarket B has a higher standard deviation, we know that there is more variation in the wait times at Supermarket B. Overall, wait times at Supermarket B are more spread out from the average whereas wait times at Supermarket A are more concentrated near the average.

• \(s=\sqrt{\frac{\Sigma {(x-\overset{\bar}{x})}^{2}}{n-1}}\) or \(s=\sqrt{\frac{\Sigma f{(x-\overset{\bar}{x})}^{2}}{n-1}}\)
• For the sample standard deviation, the denominator is n−; that is, the sample size minus 1.

• \(\sigma =\sqrt{\frac{\Sigma {(x-\mu )}^{2}}{N}}\) or \(\sigma =\sqrt{\frac{\Sigma f{(x-\mu )}^{2}}{N}}\)
• For the population standard deviation, the denominator is N, the number of items in the population.

In these formulas, f represents the frequency with which a value appears. For example, if a value appears once, f is one. If a value appears three times in the data set or population, f is three.

Condensed — the full section is in OpenStax Statistics.

Types of Variability in Samples

When researchers study a population, they often use a sample, either for convenience or because it is not possible to access the entire population. Variability is the term used to describe the differences that may occur in these outcomes. Common types of variability include the following:

• Observational or measurement variability
• Natural variability
• Induced variability
• Sample variability

Here are some examples to describe each type of variability:

Example 1: Measurement variability

Measurement variability occurs when there are differences in the instruments used to measure or in the people using those instruments. If we are gathering data on how long it takes for a ball to drop from a height by having students measure the time of the drop with a stopwatch, we may experience measurement variability if the two stopwatches used were made by different manufacturers. For example, one stopwatch measures to the nearest second, whereas the other one measures to the nearest tenth of a second. We also may experience measurement variability because two different people are gathering the data. Their reaction times in pressing the button on the stopwatch may differ; thus, the outcomes will vary accordingly. The differences in outcomes may be affected by measurement variability.

Example 2: Natural variability

Natural variability arises from the differences that naturally occur because members of a population differ from each other. For example, if we have two identical corn plants and we expose both plants to the same amount of water and sunlight, they may still grow at different rates simply because they are two different corn plants. The difference in outcomes may be explained by natural variability.

Example 3: Induced variability

Condensed — the full section is in OpenStax Statistics.

Sampling Variability of a Statistic

The statistic of a sampling distribution was discussed in Descriptive Statistics: Measures of the Center of the Data. How much the statistic varies from one sample to another is known as the sampling variability of a statistic. You typically measure the sampling variability of a statistic by its standard error. The standard error of the mean is an example of a standard error. The standard error is the standard deviation of the sampling distribution. In other words, it is the average standard deviation that results from repeated sampling. You will cover the standard error of the mean in the chapter The Central Limit Theorem (not now). The notation for the standard error of the mean is \(\frac{\sigma }{\sqrt{n}}\), where σ is the standard deviation of the population and n is the size of the sample.

Condensed — the full section is in OpenStax Statistics.

Standard deviation of Grouped Frequency Tables

Recall that for grouped data we do not know individual data values, so we cannot describe the typical value of the data with precision. In other words, we cannot find the exact mean, median, or mode. We can, however, determine the best estimate of the measures of center by finding the mean of the grouped data with the formula \(Mean\ of\ Frequency\ Table=\frac{\sum fm}{\sum f}\text{,}\)
where \(f=\) interval frequencies and m = interval midpoints.

Just as we could not find the exact mean, neither can we find the exact standard deviation. Remember that standard deviation describes numerically the expected deviation a data value has from the mean. In simple English, the standard deviation allows us to compare how unusual individual data are when compared to the mean.

Example

Find the standard deviation for the data in .

Class	Frequency, f	Midpoint, m	m2	\(\overset{\bar}{x}\)2	fm2	Standard Deviation
0–2	1	1	1	7.58	1	3.5
3–5	6	4	16	7.58	96	3.5
6–8	10	7	49	7.58	490	3.5
9–11	7	10	100	7.58	700	3.5
12–14	0	13	169	7.58	0	3.5
15–17	2	16	256	7.58	512	3.5

For this data set, we have the mean, \(\overset{\bar}{x}\) = 7.58, and the standard deviation, s_x = 3.5. This means that a randomly selected data value would be expected to be 3.5 units from the mean. If we look at the first class, we see that the class midpoint is equal to one. This is almost two full standard deviations from the mean since 7.58 – 3.5 – 3.5 = .58. While the formula for calculating the standard deviation is not complicated, \({s}_{x}=\sqrt{\frac{f{(m-\overset{\bar}{x})}^{2}}{n-1}}\text{,}\) where s_x = sample standard deviation, \(\overset{\bar}{x}\) = sample mean; the calculations are tedious. It is usually best to use technology when performing the calculations.

Comparing Values from Different Data Sets

As explained before, a z-score allows us to compare statistics from different data sets. If the data sets have different means and standard deviations, then comparing the data values directly can be misleading.

• For each data value, calculate how many standard deviations away from its mean the value is.
• In symbols, the formulas for calculating z-scores become the following.

  Sample		\(z=\frac{x\ -\ \overset{\bar}{x}}{s}\)
  Population		\(z=\frac{x\ -\ \mu }{\sigma }\)

As shown in the table, when only a sample mean and sample standard deviation are given, the top formula is used. When the population mean and population standard deviation are given, the bottom formula is used.

Example

Try it.

Two students, John and Ali, from different high schools, wanted to find out who had the highest GPA when compared to his school. Which student had the highest GPA when compared to his school?

Student	GPA	School Mean GPA	School Standard Deviation
John	2.85	3.0	.7
Ali	77	80	10

Solution

For each student, determine how many standard deviations (#ofSTDEVs) his GPA is away from the average, for his school. Pay careful attention to signs when comparing and interpreting the answer.

\(z=\# of STDEVs=\frac{\text{value }-\text{mean}}{\text{standard deviation}}=\frac{x+\mu }{\sigma }\)

For John, \(z=\#ofSTDEVs=\frac{2.85-3.0}{.7}=-0.21\)

For Ali, \(z=\#ofSTDEVs=\frac{77-80}{10}=-0.3\)

John has the better GPA when compared to his school because his GPA is 0.21 standard deviations below his school's mean, while Ali's GPA is .3 standard deviations below his school's mean.

John's z-score of –.21 is higher than Ali's z-score of –.3. For GPA, higher values are better, so we conclude that John has the better GPA when compared to his school. The z-score representing John's score does not fall as far below the mean as the z-score representing Ali's score.

The following lists give a few facts that provide a little more insight into what the standard deviation tells us about the distribution of the data.

Condensed — the full section is in OpenStax Statistics.

Measures of the Spread of the Data

For each of the examples given below, tell whether the differences in outcomes may be explained by measurement variability, natural variability, induced variability, or sampling variability.

Use the following information to answer the next two exercises. The following data are the distances between 20 retail stores and a large distribution center. The distances are in miles.
29, 37, 38, 40, 58, 67, 68, 69, 76, 86, 87, 95, 96, 96, 99, 106, 112, 127, 145, 150

പ്രവര്‍ത്തിപ്പിച്ച ഉദാഹരണം: stats 2, 4, 4, 4, 5, 5, 7, 9

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. For the following 13 real estate prices, calculate the IQR and determine if any prices are potential outliers. Prices are in dollars.
   389,950; 230,500; 158,000; 479,000; 639,000; 114,950; 5,500,000; 387,000; 659,000; 529,000; 575,000; 488,800; 1,095,000

   ഉത്തരം വെളിപ്പെടുത്തുക

   Order the following data from smallest to largest:
   114,950; 158,000; 230,500; 387,000; 389,950; 479,000; 488,800; 529,000; 575,000; 639,000; 659,000; 1,095,000; 5,500,000

   M = 488,800

   Q₁ = \(\frac{\text{230,500 + 387,000}}{2}\) = 308,750

   Q₃ = \(\frac{\text{639,000 + 659,000}}{2}\) = 649,000

   IQR = 649,000 – 308,750 = 340,250

   (1.5)(IQR) = (1.5)(340,250) = 510,375

   Q₁ – (1.5)(IQR) = 308,750 – 510,375 = –201,625

   Q₃ + (1.5)(IQR) = 649,000 + 510,375 = 1,159,375

   No house price is less than –201,625. However, 5,500,000 is more than 1,159,375. Therefore, 5,500,000 is a potential outlier.

2. For the 11 salaries, calculate the IQR and determine if any salaries are outliers. The following salaries are in dollars.

   • $33,000
   • $64,500
   • $28,000
   • $54,000
   • $72,000
   • $68,500
   • $69,000
   • $42,000
   • $54,000
   • $120,000
   • $40,500

   ഉത്തരം വെളിപ്പെടുത്തുക

   Order the data from smallest to largest:

   • $28,000
   • $33,000
   • $40,500
   • $42,000
   • $54,000
   • $54,000
   • $64,500
   • $68,500
   • $69,000
   • $72,000
   • $120,000

   Median = $54,000

   Q₁ = $40,500

   Q₃ = $69,000

   IQR = $69,000 – $40,500 = $28,500

   (1.5)(IQR) = (1.5)($28,500) = $42,750

   Q₁ – (1.5)(IQR) = $40,500 – $42,750 = –$2,250

   Q₃ + (1.5)(IQR) = $69,000 + $42,750 = $111,750

   No salary is less than –$2,250. However, $120,000 is more than $11,750, so $120,000 is a potential outlier.

3. Find the interquartile range for the following two data sets and compare them.

   Test Scores for Class A:
   69, 96, 81, 79, 65, 76, 83, 99, 89, 67, 90, 77, 85, 98, 66, 91, 77, 69, 80, 94
   Test Scores for Class B:
   90, 72, 80, 92, 90, 97, 92, 75, 79, 68, 70, 80, 99, 95, 78, 73, 71, 68, 95, 100

   ഉത്തരം വെളിപ്പെടുത്തുക

   Class A

   Order the data from smallest to largest:

   65, 66, 67, 69, 69, 76, 77, 77, 79, 80, 81, 83, 85, 89, 90, 91, 94, 96, 98, 99

   \(\text{Median}=\frac{80+81}{2}=80.5\)

   \({Q}_{1}=\frac{69+76}{2}=72.5\)

   \({Q}_{3}=\frac{90+91}{2}=90.5\)

   IQR = 90.5 – 72.5 = 18

   Class B

   Order the data from smallest to largest:

   68, 68, 70, 71, 72, 73, 75, 78, 79, 80, 80, 90, 90, 92, 92, 95, 95, 97, 99, 100

   \(\text{Median}=\frac{80+80}{2}=80\)

   \({Q}_{1}=\frac{72+73}{2}=72.5\)

   \({Q}_{3}=\frac{92+95}{2}=93.5\)

   IQR = 93.5 – 72.5 = 21

   The data for Class B has a larger IQR, so the scores between Q₃ and Q₁ (middle 50%) for the data for Class B are more spread out and not clustered about the median.

4. Forty bus drivers were asked how many hours they spend each day running their routes (rounded to the nearest hour). Find the 65^th percentile.

   Amount of Time Spent on Route (Hours)	Frequency	Relative Frequency	Cumulative Relative Frequency
   2	12	.30	.30
   3	14	.35	.65
   4	10	.25	.90
   5	4	.10	1.00

   ഉത്തരം വെളിപ്പെടുത്തുക

   The 65^th percentile is between the last three and the first four.

   The 65^th percentile is 3.5.

5. Using :

   1. Find the 80^th percentile.
   2. Find the 90^th percentile.
   3. Find the first quartile. What is another name for the first quartile?
   ഉത്തരം വെളിപ്പെടുത്തുക

   Using the data from the frequency table, we have the following:

   1. The 80^th percentile is between the last eight and the first nine in the table (between the 40^th and 41^st values). Therefore, we need to take the mean of the 40^th an 41^st values. The 80^th percentile \(=\frac{8+9}{2}=8.5\text{.}\)
   2. The 90^th percentile will be the 45^th data value (location is 0.90(50) = 45), and the 45^th data value is nine.
   3. Q₁ is also the 25^th percentile. The 25^th percentile location calculation: P₂₅ = .25(50) = 12.5 ≈ 13, the 13^th data value. Thus, the 25^th percentile is six.

6. Refer to . Find the third quartile. What is another name for the third quartile?

   ഉത്തരം വെളിപ്പെടുത്തുക

   The third quartile is the 75^th percentile, which is four. The 65^th percentile is between three and four, and the 90^th percentile is between four and 5.75. The third quartile is between 65 and 90, so it must be four.

7. Listed are 29 ages for Academy Award-winning best actors in order from smallest to largest:
   18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77

   1. Find the 70^th percentile.
   2. Find the 83^rd percentile.
   ഉത്തരം വെളിപ്പെടുത്തുക

   1. • k = 70
      • i = the index
      • n = 29
      i = \(\frac{k}{100}\) (n + 1) = (\(\frac{70}{100}\))(29 + 1) = 21. This equation tells us that i, or the position of the data value in the data set, is 21. So, we will count over to the 21^st position, which shows a data value of 64.
   2. • k = 83^rd percentile
      • i = the index
      • n = 29
      i  = \(\frac{k}{100}\) (n + 1) = (\(\frac{83}{100}\))(29 + 1) = 24.9, which is not an integer. Round it down to 24 and up to 25. The age in the 24^th position is 71, and the age in the 25^th position is 72. Average 71 and 72. The 83^rd percentile is 71.5 years.

8. Listed are 29 ages for Academy Award-winning best actors in order from smallest to largest:

   18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77
   Calculate the 20^th percentile and the 55^th percentile.

   ഉത്തരം വെളിപ്പെടുത്തുക

   k = 20. Index = i = \(\frac{k}{100}(n+1)=\frac{20}{100}\)(29 + 1) = 6. The age in the sixth position is 27. The 20^th percentile is 27 years.

   k = 55. Index = i = \(\frac{k}{100}(n+1)=\frac{55}{100}\)(29 + 1) = 16.5. Round down to 16 and up to 17. The age in the 16^th position is 52 and the age in the 17^th position is 55. The average of 52 and 55 is 53.5. The 55^th percentile is 53.5 years.

9. Listed are 29 ages for Academy Award-winning best actors in order from smallest to largest:
   18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77

   1. Find the percentile for 58.
   2. Find the percentile for 25.
   ഉത്തരം വെളിപ്പെടുത്തുക

   1. Counting from the bottom of the list, there are 18 data values less than 58. There is one value of 58.

      x = 18 and y = 1.\(\frac{x+.5y}{n}\)(100) = \(\frac{18+.5(1)}{29}\)(100) = 63.80. Fifty-eight is the 64^th percentile.

   2. Counting from the bottom of the list, there are three data values less than 25. There is one value of 25.

      x = 3 and y = 1.\(\frac{x+.5y}{n}\)(100) = \(\frac{3+.5(1)}{29}\)(100) = 12.07. Twenty-five is the 12^th percentile.

10. Listed are 30 ages for Academy Award-winning best actors in order from smallest to largest:

    18, 21, 22, 25, 26, 27, 29, 30, 31, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77
    Find the percentiles for 47 and 31.

    ഉത്തരം വെളിപ്പെടുത്തുക

    Percentile for 47: Counting from the bottom of the list, there are 15 data values less than 47. There is one value of 47.

    x = 15 and y = 1.\(\frac{x+0.5y}{n}\)(100) = \(\frac{15+0.5(1)}{29}\)(100) = 53.45. 47 is the 53^rd percentile.

    Percentile for 31: Counting from the bottom of the list, there are eight data values less than 31. There are two values of 31.

    x = 15 and y = 2.\(\frac{x+0.5y}{n}\)(100) = \(\frac{15+0.5(2)}{29}\)(100) = 31.03. 31 is the 31^st percentile.

11. On a timed math test, the first quartile for time it took to finish the exam was 35 minutes. Interpret the first quartile in the context of this situation.

    ഉത്തരം വെളിപ്പെടുത്തുക

    • Twenty-five percent of students finished the exam in 35 minutes or less.
    • Seventy-five percent of students finished the exam in 35 minutes or more.
    • A low percentile could be considered good, as finishing more quickly on a timed exam is desirable. If you take too long, you might not be able to finish.

12. For the 100-meter dash, the third quartile for times for finishing the race was 11.5 seconds. Interpret the third quartile in the context of the situation.

    ഉത്തരം വെളിപ്പെടുത്തുക

    Twenty-five percent of runners finished the race in 11.5 seconds or more. Seventy-five percent of runners finished the race in 11.5 seconds or less. A lower percentile is good because finishing a race more quickly is desirable.

13. On a 20-question math test, the 70^th percentile for number of correct answers was 16. Interpret the 70^th percentile in the context of this situation.

    ഉത്തരം വെളിപ്പെടുത്തുക

    • Seventy percent of students answered 16 or fewer questions correctly.
    • Thirty percent of students answered 16 or more questions correctly.
    • A higher percentile could be considered good, as answering more questions correctly is desirable.

14. On a 60-point written assignment, the 80^th percentile for the number of points earned was 49. Interpret the 80^th percentile in the context of this situation.

    ഉത്തരം വെളിപ്പെടുത്തുക

    Eighty percent of students earned 49 points or fewer. Twenty percent of students earned 49 or more points. A higher percentile is good because getting more points on an assignment is desirable.

15. At a high school, it was found that the 30^th percentile of number of hours that students spend studying per week is seven hours. Interpret the 30^th percentile in the context of this situation.

    ഉത്തരം വെളിപ്പെടുത്തുക

    • Thirty percent of students study seven or fewer hours per week.
    • Seventy percent of students study seven or more hours per week.
    • In this example, there is not necessarily a good or bad value judgment associated with a higher or lower percentile, since the time a student studies per week is dependent on his/her needs.

16. During a season, the 40^th percentile for points scored per player in a game is eight. Interpret the 40^th percentile in the context of this situation.

    ഉത്തരം വെളിപ്പെടുത്തുക

    Forty percent of players scored eight points or fewer. Sixty percent of players scored eight points or more. A higher percentile is good because getting more points in a basketball game is desirable.

17. Listed are 29 ages for Academy Award-winning best actors in order from smallest to largest:

    18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77

    1. Find the 40^th percentile.
    2. Find the 78^th percentile.
    ഉത്തരം വെളിപ്പെടുത്തുക

    1. The 40^th percentile is 37 years.
    2. The 78^th percentile is 70 years.

18. Listed are 32 ages for Academy Award-winning best actors in order from smallest to largest:

    18, 18, 21, 22, 25, 26, 27, 29, 30, 31, 31, 33, 36, 37, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77

    1. Find the percentile of 37.
    2. Find the percentile of 72.

19. Jesse was ranked 37^th in his graduating class of 180 students. At what percentile is Jesse’s ranking?

    ഉത്തരം വെളിപ്പെടുത്തുക

    Jesse graduated 37^th out of a class of 180 students. There are 180 – 37 = 143 students ranked below Jesse. There is one rank of 37.

    x = 143 and y = 1. \(\frac{x+.5y}{n}\)(100) = \(\frac{143+.5(1)}{180}\)(100) = 79.72. Jesse’s rank of 37 puts him at the 80^th percentile.

20. 1. For runners in a race, a low time means a faster run. The winners in a race have the shortest running times. Is it more desirable to have a finish time with a high or a low percentile when running a race?
    2. The 20^th percentile of run times in a particular race is 5.2 minutes. Write a sentence interpreting the 20^th percentile in the context of the situation.
    3. A bicyclist in the 90^th percentile of a bicycle race completed the race in 1 hour and 12 minutes. Is he among the fastest or slowest cyclists in the race? Write a sentence interpreting the 90^th percentile in the context of the situation.
21. 1. For runners in a race, a higher speed means a faster run. Is it more desirable to have a speed with a high or a low percentile when running a race?
    2. The 40^th percentile of speeds in a particular race is 7.5 miles per hour. Write a sentence interpreting the 40^th percentile in the context of the situation.
    ഉത്തരം വെളിപ്പെടുത്തുക

    1. For runners in a race, it is more desirable to have a high percentile for speed. A high percentile means a higher speed, which is faster.
    2. 40 percent of runners ran at speeds of 7.5 miles per hour or less (slower), and 60 percent of runners ran at speeds of 7.5 miles per hour or more (faster).

22. On an exam, would it be more desirable to earn a grade with a high or a low percentile? Explain.

23. Mina is waiting in line at the Department of Motor Vehicles. Her wait time of 32 minutes is the 85^th percentile of wait times. Is that good or bad? Write a sentence interpreting the 85^th percentile in the context of this situation.

    ഉത്തരം വെളിപ്പെടുത്തുക

    When waiting in line at the DMV, the 85^th percentile would be a long wait time compared to the other people waiting. 85 percent of people had shorter wait times than Mina. In this context, Mina would prefer a wait time corresponding to a lower percentile. 85 percent of people at the DMV waited 32 minutes or less. 15 percent of people at the DMV waited 32 minutes or longer.

24. In a survey collecting data about the salaries earned by recent college graduates, Li found that her salary was in the 78^th percentile. Should Li be pleased or upset by this result? Explain.

25. In a study collecting data about the repair costs of damage to automobiles in a certain type of crash tests, a certain model of car had $1,700 in damage and was in the 90^th percentile. Should the manufacturer and the consumer be pleased or upset by this result? Explain and write a sentence that interprets the 90^th percentile in the context of this problem.

    ഉത്തരം വെളിപ്പെടുത്തുക

    The manufacturer and the consumer would be upset. This is a large repair cost for the damages, compared to the other cars in the sample. INTERPRETATION: 90 percent of the crash-tested cars had damage repair costs of $1,700 or less; only 10 percent had damage repair costs of $1,700 or more.

26. The University of California has two criteria used to set admission standards for freshman to be admitted to a college in the UC system:

    1. Students' GPAs and scores on standardized tests (SATs and ACTs) are entered into a formula that calculates an admissions index score. The admissions index score is used to set eligibility standards intended to meet the goal of admitting the top 12 percent of high school students in the state. In this context, what percentile does the top 12 percent represent?
    2. Students whose GPAs are at or above the 96^th percentile of all students at their high school are eligible, called eligible in the local context, even if they are not in the top 12 percent of all students in the state. What percentage of students from each high school are eligible in the local context?

27. Suppose that you are buying a house. You and your real estate agent have determined that the most expensive house you can afford is the 34^th percentile. The 34^th percentile of housing prices is $240,000 in the town you want to move to. In this town, can you afford 34 percent of the houses or 66 percent of the houses?

    ഉത്തരം വെളിപ്പെടുത്തുക

    You can afford 34 percent of houses. 66 percent of the houses are too expensive for your budget. INTERPRETATION: 34 percent of houses cost $240,000 or less; 66 percent of houses cost $240,000 or more.

28. First quartile = ________

29. Second quartile = median = 50^th percentile = ________

    ഉത്തരം വെളിപ്പെടുത്തുക

    4

30. Third quartile = ________

31. Interquartile range (IQR) = ________ – ________ = ________

    ഉത്തരം വെളിപ്പെടുത്തുക

    6 – 4 = 2

32. 10^th percentile = ________

33. 70^th percentile = ________

    ഉത്തരം വെളിപ്പെടുത്തുക

    6

34. The median age for U.S. ethnicity A currently is 30.9 years; for U.S. ethnicity B, it is 42.3 years.

    1. Based on this information, give two reasons why ethnicity A median age could be lower than the ethnicity B median age.
    2. Does the lower median age for ethnicity A necessarily mean that ethnicity A die younger than ethnicity B? Why or why not?
    3. How might it be possible for ethnicity A and ethnicity B to die at approximately the same age but for the median age for ethnicity B to be higher?

35. Six hundred adult Americans were asked by telephone poll, "What do you think constitutes a middle-class income?" The results are in . Also, include the left endpoint but not the right endpoint.

    Salary ($)	Relative Frequency
    < 20,000	.02
    20,000–25,000	.09
    25,000–30,000	.19
    30,000–40,000	.26
    40,000–50,000	.18
    50,000–75,000	.17
    75,000–99,999	.02
    100,000+	.01

    1. What percentage of the survey answered "not sure"?
    2. What percentage think that middle class is from $25,000 to $50,000?
    3. Construct a histogram of the data.
       1. Should all bars have the same width, based on the data? Why or why not?
       2. How should the < 20,000 and the 100,000+ intervals be handled? Why?
    4. Find the 40^th and 80^th percentiles.
    5. Construct a bar graph of the data.
    ഉത്തരം വെളിപ്പെടുത്തുക

    1. 1 – (.02+.09+.19+.26+.18+.17+.02+.01) = .06
    2. .19+.26+.18 = .63
    3. Check student’s solution.

    4. 40^th percentile will fall between 30,000 and 40,000

       80^th percentile will fall between 50,000 and 75,000

    5. Check student’s solution.

36. Given the following box plot, answer the questions.

    1. Which quarter has the smallest spread of data? What is that spread?
    2. Which quarter has the largest spread of data? What is that spread?
    3. Find the interquartile range (IQR).
    4. Are there more data in the interval 5–10 or in the interval 10–13? How do you know this?
    5. Which interval has the fewest data in it? How do you know this?
       1. 0–2
       2. 2–4
       3. 10–12
       4. 12–13
       5. need more information

37. The following box plot shows the ages of the U.S. population for 1990, the latest available year:

    1. Are there fewer or more children (age 17 and under) than senior citizens (age 65 and over)? How do you know?
    2. 12.6 percent are age 65 and over. Approximately what percentage of the population are working-age adults (above age 17 to age 65)?
    ഉത്തരം വെളിപ്പെടുത്തുക

    1. more children; the left whisker shows that 25 percent of the population are children 17 and younger; the right whisker shows that 25 percent of the population are adults 50 and older, so adults 65 and over represent less than 25 percent
    2. 62.4 percent

38. Data indicating the number of months a patient with a specific disease lives after taking a new antibody drug are as follows (smallest to largest):
    3, 4, 8, 8, 10, 11, 12, 13, 14, 15, 15, 16, 16, 17, 17, 18, 21, 22, 22, 24, 24, 25, 26, 26, 27, 27, 29, 29, 31, 32, 33, 33, 34, 34, 35, 37, 40, 44, 44, 47
    Calculate the mean and the median.

    ഉത്തരം വെളിപ്പെടുത്തുക

    The calculation for the mean is

    \(\begin{array}{l}\overset{\bar}{x}=[3+4+(8)(2)+10+11+12+13+14+(15)(2)+(16)(2)+(17)(2)+18+21+(22)(2)+(24)(2)+25+(26)(2) \\ +(27)(2)+(29)(2)+31+32+(33)(2)+(34)(2)+35+37+40+(44)(2)+47]/40=23.6.\end{array}\)
    To find the median, M, first use the formula for the location. The location is
    \(\frac{n+1}{2}=\frac{40+1}{2}=20.5.\)
    Start from the smallest value and count up; the median is located between the 20^th and 21^st values (the two 24s):
    3, 4, 8, 8, 10, 11, 12, 13, 14, 15, 15, 16, 16, 17, 17, 18, 21, 22, 22, 24, 24, 25, 26, 26, 27, 27, 29, 29, 31, 32, 33, 33, 34, 34, 35, 37, 40, 44, 44, 47

    \(M=\frac{24+24}{2}=24\)

39. The following data show the number of months patients typically wait on a transplant list before getting surgery. The data are ordered from smallest to largest. Calculate the mean and median.

    3, 4, 5, 7, 7, 7, 7, 8, 8, 9, 9, 10, 10, 10, 10, 10, 11, 12, 12, 13, 14, 14, 15, 15, 17, 17, 18, 19, 19, 19, 21, 21, 22, 22, 23, 24, 24, 24, 24

    ഉത്തരം വെളിപ്പെടുത്തുക

    Mean: 3 + 4 + 5 + 7 + 7 + 7 + 7 + 8 + 8 + 9 + 9 + 10 + 10 + 10 + 10 + 10 + 11 + 12 + 12 + 13 + 14 + 14 + 15 + 15 + 17 + 17 + 18 + 19 + 19 + 19 + 21 + 21 + 22 + 22 + 23 + 24 + 24 + 24 = 544
    \(\frac{\text{544}}{\text{39}}\text{ = 13}\text{.95}\)
    Median: Starting at the smallest value, the median is the 20^th term, which is 13.

40. Suppose that in a small town of 50 people, one person earns $5,000,000 per year and the other 49 each earn $30,000. Which is the better measure of the center: the mean or the median?

    ഉത്തരം വെളിപ്പെടുത്തുക

    \(\overset{\bar}{x}=\frac{5,000,000+49(30,000)}{50}=129,400\)

    M = 30,000

    There are 49 people who earn $30,000 and one person who earns $5,000,000.

    The median is a better measure of the center than the mean because 49 of the values are 30,000 and one is 5,000,000. The 5,000,000 is an outlier. The 30,000 gives us a better sense of the middle of the data.

നീ സ്വയം ശ്രമിക്ക്.

⌨ Type ✎ Draw

നിങ്ങളുടെ കണക്കുകൂട്ടലുകൾ

പരിഹരിക്കുക

x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

mean of 10, 20, 30, 45median of 3, 1, 4, 1, 5, 9, 2, 6standard deviation of 2, 4, 4, 4, 5, 5, 7, 9variance of 1, 2, 3, 4, 5

Parts of this page are adapted from OpenStax Statistics (CC BY 4.0). Condensed and re-explained here; errors are ours.

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