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- Understanding normal distributions
- Stats and R
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- Lesson 4: Bell-Shaped Curves and Statistical Pictures
Understanding normal distributions
In Lesson 4 we continue our discussion of describing data through numerical summaries and also think about statistical pictures. An overview of some key questions addressed is given in the table below. In Lesson 3 we learned that the standard deviation provides a measure of variability about the mean. Generally, most observations are within one standard deviation of the mean, and observations more than three standard deviations away from the mean are very rare.
Thus, the standard deviation provides a natural yardstick by which to gauge where an observation stands relative to others. If you are five standard deviations above the mean then you know you are at the top of the list; one standard deviation below the mean and you know you are on the low side but not too far down.
To compute the standardized score of a value, you take. These numbers are called "standardized" because the list of standardized scores itself always has a mean of 0 and a standard deviation of 1.
That's because subtracting the mean from every value makes the new mean equal zero and dividing every value by the standard deviation makes the new standard deviation equal to 1. According to EPA data, the gas mileage for compact SUVs in the model year has a mean of approximately 22 mpg and a standard deviation of about 3 mpg. One SUV gets 25 mpg. It is one standard deviation above the mean.
Another SUV gets It is one-half of a standard deviation below the mean. The standardized scores give you a way to compare relative standing of values on different lists where the distributions might have roughly similar shapes. According to EPA data, 4-cylinder model year cars have CO2 emissions that average ppm parts per million with a standard deviation of 51 ppm; while 6-cylinder cars made that year average ppm with a standard deviation of 44 ppm.
Which vehicle has higher CO2 emissions relative to other cars with the same number of cylinders; the 4-cylinder Honda Civic that emits ppm or the 6-cylinder Toyota Camry that emits ppm? Many measurement variables found in nature follow a predictable pattern.
The predictable pattern of interest is a type of symmetry where much of the distribution of the data is clumped around the center and few observations are found on the extremes.
Data that has this pattern are said to be bell-shaped or have a normal distribution. It can be shown that variables that arise as a result of the sum or average of a fixed number of individual smaller components of a similar nature will have this shape. Thus, the distribution of the weights of cartons of large eggs at a grocery store will look like a normal curve because the weight of a carton arises from the sum of the weights of the dozen eggs inside.
Many measures used by psychologists to gauge levels of characteristics like stress or anxiety or happiness are based on questionnaires that score your answers to lots of individual questions and then sum them up to get a final measure. The distributions of such measures within a homogeneous group of people will then approximately follow a normal curve.
Figure 4. Since a normal distribution is a type of symmetric distribution, you would expect the mean and median to be very close in value. With this example, the mean is Notice the upper tail where the data is clumped. However, the mean and median are still pretty close, and using the normal curve to calculate percentiles for example should give very rough approximations. It is likely that the GPA variable would look more like a normal curve if the data were restricted to a more homogeneous group with a similar number of credit hours taken.
The major problem with this variable is that it is extremely skewed to the right since most people have no tattoos at all. Also, the graph has gaps because this variable is discrete with only a few values in the data set. Thus, the normal curve should not be used to make even rough approximations for data about the number of tattoos.
Since the histogram shows that this data is normally distributed, the empirical rule can be applied. The mean and standard deviation SD for this sample is Below are the calculations for the sample of heights. One would expect it to be very unusual for someone in this sample to be smaller than An important feature of the normal curve is that percentiles are completely determined by the standardized scores. Table 8. The mean mileage was What percentage of the compact SUVs got worse mileage than the Encore?
Next, we look at Table 8. In this section, we examine a few important types of statistical pictures: bar graphs, time series plots, and scatterplots. Before turning to these specific types of statistical pictures, it is important to note that regardless of the type of picture being used, there are some basic features that a good graph will possess:.
Section 9. The Gallup World Poll takes random samples of the adults in different countries. In many of those countries, Gallup asks respondents to try and think about an overall evaluation of their lives and to specify how satisfied they are on a four-point scale very dissatisfied, dissatisfied, satisfied, or very satisfied. Bar graphs are often used to show the results of data for categorical variables and, as in Figure 4. Each day the Gallup Poll takes a random sample of about American adults nationally and asks them about a variety of issues including how much money they spent the day before not counting the purchase of a home or car or paying normal household bills like for electrical or phone service.
For example, each point on the dark green line represents the average results of the amount spent by the American adults who had responded to the survey over the 3-day period leading up to the day of the survey. This type of line graph is called a time series plot because the points represent the variable being measured across time.
When looking at a time series plot like Figure 4. An experiment was carried out to see how Blood Alcohol Content BAC as measured by a breathalyzer change with the number of ounce beers you drink the experiment is discussed in the Electronic Encyclopedia of Statistics Examples and Exercises.
In the experiment, 16 subjects each drew a number out of a hat. For example, if the number was a 3, then that subject drank 3 beers. A half-hour after finishing the last assigned beer a police officer used a breathalyzer, like the ones they use in the field, to measure the subject's BAC level.
Each point represents a different subject. For example, one subject drank 6 beers and had a BAC of 0. Scatterplots are used for displaying the relationship between two measurement variables. Examining Figure 4. The data here were based on a randomized experiment and the causal nature of this particular relationship is quite well established. Positive associations are reflected in a cloud of points in the scatterplot that goes "uphill" as you move from left to right. A negative association , like the one we would see if we plotted the weights of cars versus their gas mileage, shows a cloud of points going "downhill".
Click the right arrow to proceed to the next question. When you have completed all of the questions you will see how many you got right and the correct answers. Your browser does not support the audio element. Lesson Overview In Lesson 4 we continue our discussion of describing data through numerical summaries and also think about statistical pictures. Question Addressed Statistical Summary Where are values located along the number line?
Median Mean How variable are the numbers? IQR Standard Deviation What is the relative standing of an individual value compared with other numbers on a list? Dot plots Boxplots Histograms normal or bell-shaped curve is a special case How is a categorical variable distributed? How do categorical variables compare? Bar graphs Comparative bar graphs How do the distributions of two measurement variables compare? Comparative boxplots Comparative dotplots How do percentages or averages change over time?
Line graph or time series plot How are two measurement variables associated? Objectives After successfully completing this lesson, you should be able to:. Standardized Scores also called "standard scores" or "z-scores" In Lesson 3 we learned that the standard deviation provides a measure of variability about the mean. Example 4. The Honda Civic has a higher relative CO2 emissions. Before turning to these specific types of statistical pictures, it is important to note that regardless of the type of picture being used, there are some basic features that a good graph will possess: The data should be clearly recognizable from the background The picture should be clearly labeled, showing the title and purpose or origin of the data, what is being plotted on each axis, bar, or segment of the plot i.
For example, in the consumer spending data, we can see a long-term generally upward trend since the end of the recession in June One note of caution when looking at economic data that extends over decades in time; check if they have been adjusted for inflation. An apparent upward trend may be nothing more than reflecting a change in the value of the dollar. Are there seasonal components? While temperature data is dramatically affected by regular seasonal cycles, many other variables change in predictable patterns because of people's behavioral changes in certain months or seasons.
For example, have a close look at Figure 4. You should be able to see a bump in consumer spending each year associated with the holidays in December. There are other cyclic effects in this data. If you look really closely at the 3-day averages, you can see that there is increased spending on weekends compared with weekdays.
What is the nature of the random fluctuations? We know that every measurement is subject to natural variability and that averages will be more reliable if they are based on larger sample sizes. Have a look at Figure 4. Think About It! Select the answer you think is correct - then click the 'Check' button to see how you did. Have Fun With It! Dot plots Boxplots Histograms normal or bell-shaped curve is a special case.
How is a categorical variable distributed?
Stats and R
The normal distribution is a function that defines how a set of measurements is distributed around the center of these measurements i. Many natural phenomena in real life can be approximated by a bell-shaped frequency distribution known as the normal distribution or the Gaussian distribution. The normal distribution is a mount-shaped, unimodal and symmetric distribution where most measurements gather around the mean. Moreover, the further a measure deviates from the mean, the lower the probability of occurring. In this sense, for a given variable, it is common to find values close to the mean, but less and less likely to find values as we move away from the mean. Last but not least, since the normal distribution is symmetric around its mean, extreme values in both tails of the distribution are equivalently unlikely. For instance, given that adult height follows a normal distribution, most adults are close to the average height and extremely short adults occur as infrequently as extremely tall adults.
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When examining data, it is often best to create a graphical representation of the distribution. Visual graphs, such as histograms, help one to easily see a few very important characteristics about the data, such as its overall pattern, striking deviations from that pattern, and its shape, center, and spread. A histogram is particularly useful when there is a large number of observations. Histograms break the range of values in classes, and display only the count or percent of the observations that fall into each class. This chapter will focus specifically on probability histograms, which is an idealization of the relative frequency distribution.
Published on October 23, by Pritha Bhandari. Revised on January 19, In a normal distribution, data is symmetrically distributed with no skew. When plotted on a graph, the data follows a bell shape, with most values clustering around a central region and tapering off as they go further away from the center. Normal distributions are also called Gaussian distributions or bell curves because of their shape.
This lab discusses the basics of visualizing data, probability, the normal distribution, and z scores. The following packages are required for this lab:. Recall that histograms are used to visualize continuous data.
Documentation Help Center. The normal distribution, sometimes called the Gaussian distribution, is a two-parameter family of curves. The usual justification for using the normal distribution for modeling is the Central Limit theorem, which states roughly that the sum of independent samples from any distribution with finite mean and variance converges to the normal distribution as the sample size goes to infinity.
Lesson 4: Bell-Shaped Curves and Statistical Pictures
In Lesson 4 we continue our discussion of describing data through numerical summaries and also think about statistical pictures. An overview of some key questions addressed is given in the table below. In Lesson 3 we learned that the standard deviation provides a measure of variability about the mean. Generally, most observations are within one standard deviation of the mean, and observations more than three standard deviations away from the mean are very rare. Thus, the standard deviation provides a natural yardstick by which to gauge where an observation stands relative to others. If you are five standard deviations above the mean then you know you are at the top of the list; one standard deviation below the mean and you know you are on the low side but not too far down. To compute the standardized score of a value, you take.
The shape of the bar graphs and histograms of almost any kind of numerical data resembles a Bell-shaped pattern. A perfect such bell-shape is shown in the following diagram: Picture Example A : see Page The following is the Frequency table of the scores obtained by a group of students:. Example B : see Page Following is the class frequency table of weight in pounds of some babies:. Class Frequency 15 24 41 67 26 5 2.
В воздухе ощущался едва уловимый запах озона. Остановившись у края люка, Сьюзан посмотрела. Фреоновые вентиляторы с урчанием наполняли подсобку красным туманом. Прислушавшись к пронзительному звуку генераторов, Сьюзан поняла, что включилось аварийное питание. Сквозь туман она увидела Стратмора, который стоял внизу, на платформе. Прислонившись к перилам, он вглядывался в грохочущее нутро шахты ТРАНСТЕКСТА.
Вы на чуть-чуть опоздали. - Ее слова словно повисли в воздухе. Все-таки он опоздал. Плечи Беккера обмякли. - А на этот рейс были свободные места. - Сколько угодно, - улыбнулась женщина. - Самолет улетел почти пустой.
Совершенно. Будет очень глупо, если вы этого не сделаете. На этот раз Стратмор позволил себе расхохотаться во весь голос. - Твой сценарий мне понятен. ТРАНСТЕКСТ перегрелся, поэтому откройте двери и отпустите. - Именно так, черт возьми. Я был там, внизу.
Уже на середине комнаты она основательно разогналась. За полтора метра до стеклянной двери Сьюзан отпрянула в сторону и зажмурилась. Раздался страшный треск, и стеклянная панель обдала ее дождем осколков. Звуки шифровалки впервые за всю историю этого здания ворвались в помещение Третьего узла. Сьюзан открыла .
Это приказ. Чатрукьян замер от неожиданности. - Но, сэр, мутация… - Немедленно! - крикнул Стратмор.
- Я мог бы предложить вам более привлекательную идею. - Ролдан был человек осторожный, а визит в полицию мог превратить его клиентов в бывших клиентов. - Подумайте, - предложил .
Купол здания, похожий на спутник, находился в ста девяти ярдах от основного здания АНБ, и попасть туда можно было только через главный вход. Поскольку в шифровалке имелось автономное энергоснабжение, на главный распределительный щит, наверное, даже не поступил сигнал, что здесь произошла авария. - Основное энергоснабжение вырубилось, - сказал Стратмор, возникший за спиной Сьюзан.