Scatter plots can be effective in measuring the strength â¦ In other words, each individual (driver, in our example) appears on the scatterplot as a single point whose X-coordinate is the value of the explanatory variable for that individual, and whose Y-coordinate is the value of the response variable. We will discuss that later in this section. Notes. In general terms, by looking at the scatterplot we can estimate the strength â¦ On the other hand, the gestation periods of animals that live 12 years vary much more, and range from about 60 days up to more than 400 days. An arrow drawn over the scatterplot illustrates the negative direction of this relationship: The form of the relationship seems to be linear. Recall that the example examined how the percentage of participants who completed a survey is affected by the monetary incentive that researchers promised to participants. If you have found these materials helpful, DONATE by clicking on the "MAKE A GIFT" link below or at the top of the page! In statistics, the correlation coefficient r measures the strength and direction of a linear relationship between two variables on a scatterplot. Workshop statistics: Discovery with data and Minitab. The strength is determined by the numerical value of the correlation. The first step in exploring the relationship between driver age and sign legibility distance is to create an appropriate and informative graphical display. Assessing the strength just by looking at the scatterplot can be problematic; using a numerical measure to determine strength is discussed later in this course. ... To determine the strength â¦ Although, as we mentioned earlier, it is problematic to assess the strength without a numerical measure, the relationship appears to be moderately strong, as the data is fairly tightly scattered about the line. In the right scatterplot, the points also follow the linear pattern, but much less closely, and therefore we can say that the relationship is weaker. It appears that there is a positive relationship within all three types. It's important to note that scatter plots show correlation between two variables, from which causation only may be inferred. A scatter plot identifies a possible relationship between changes observed in two different sets of variables. The form displays the phenomenon of “diminishing returns” — a return rate that after a certain point fails to increase proportionately to additional outlays of investment. Another feature of the scatterplot that is worth observing is how the variation in gestation increases as longevity increases. Outliers in scatter plots. Which scatterplot has a correlation coefficient of 0.99? It provides a visual and statistical means to test the strength of a relationship between two variables. Sources. Adding labels to the scatterplot that indicate â¦ Bivariate relationship linearity, strength and direction. ; Fundamentally, scatter works with 1-D arrays; x, y, s, and c may be input as 2-D arrays, but within scatter â¦ In the previous two cases we had a categorical explanatory variable, and therefore exploring the relationship between the two variables was done by comparing the distribution of the response variable for each category of the explanatory variable: Case Q→Q is different in the sense that both variables (in particular the explanatory variable) are quantitative. A scatter plot identifies a possible relationship between changes observed in two different sets of variables. It can be somewhat subjective to compare the strength of one association to another. Scatter plots are particularly helpful graphs when we want to see if there is a linear relationship among data points. Introduction to the practice of statistics. Not all relationships can be classified as either positive or negative. The result is sometimes called a labeled scatterplot or grouped scatterplot, and can provide further insight about the relationship we are exploring. Have direction, form and strength. A positive (or increasing) relationship means that an increase in one of the variables is associated with an increase in the other. Here is an example. A. Scatterplot A. Since the purpose of this study is to explore the effect of age on maximum legibility distance. (1985). The Department of Biostatistics will use funds generated by this Educational Enhancement Fund specifically towards biostatistics education. A line of best fit is used in the scatter plot to assess the strength or weakness of a linear relationship. This is true whether the pattern is linear, nonlinear, positive, or negative. More precise evidence is needed, and this evidence is obtained by computing a coefficient that measures the strength â¦ This scatter plot from The Atlantic Cities (2012) plots a city's "Metro Health Index" (a factor measuring the share of people who smoke or are obese) as it correlates to the city's median income. For scatterplots with linear patterns, the correlation coefficient can be usedto better understand this strength. Health Care Facilities, Providers, and Insurance, Healthy Communities, Environment and Workplaces, Resource Library for Advancing Health Equity, Contact the Center for Public Health Practice, Miller, Moore, Richards, and McKaig (PDF). This means that it is a map of two variables (typically labeled as X and Y) that are paired with each other. ... Bivariate relationship linearity, strength and direction. An arrow drawn over the scatterplot below illustrates this: The form of the relationship is again essentially linear. 111). This forms a non-linear (curvilinear) relationship that seems to be very strong, as the observations seem to perfectly fit the curve. Practice: Positive and negative linear associations from scatter plots. This indicates how strong in your memory this concept is. The relationship between two quantitative variables is visually displayed using the, When we explore a relationship using the scatterplot we should describe the. The scatterplot below displays the relationship between the sodium and calorie content of 54 brands of hot dogs. Data on the average gestation period and longevity (in captivity) of 40 different species of animals have been examined, with the purpose of examining how the gestation period of an animal is related to (or can be predicted from) its longevity. The strength of the relationship is determined by how closely the data points follow the form. Practice: Describing trends in scatter plots. Lam. This can provide an additional signal as to how strong â¦ Scatter plots can be effective in measuring the strength of relationships uncovered with a fishbone diagram. Maybe if we label the scatterplot, indicating the type of hot dogs, we will get a better understanding of the form. They indicate both the direction of the relationship between the x variables and the y variables, and the strength â¦ The strength of a correlation indicates how strong the relationship is between the two variables. In certain circumstances, it may be reasonable to indicate different subgroups or categories within the data on the scatterplot, by labeling each subgroup differently. The value of r is always between +1 and â1. The following figure summarizes this point: As the figure explains, when describing the overall pattern of the relationship we look at its direction, form and strength. For example, suppose you want to show the pattern of accidents happening on the â¦ Practice. Public Health Foundation, GOAL/QPC, 651-201-5000 Phone Core (Data Analysis) Tutorial 17: Interpreting Scatterplots. Scatter plots are a good way to predict and determine the nature of an unknown variable by plotting it with a known one. Plot points and estimate the line that best represents them % Progress . (Reference: Utts and Heckard, Mind on Statistics (2002). Here is how a scatterplot is constructed for our example: To create a scatterplot, each pair of values is plotted, so that the value of the explanatory variable (X) is plotted on the horizontal axis, and the value of the response variable (Y) is plotted on the vertical axis. The plot function will be faster for scatterplots where markers don't vary in size or color. We can see that in the left scatterplot the data points follow the linear pattern quite closely. The example in the last activity provides a great opportunity for interpretation of the form of the relationship in context. You can identify basic patterns using a scatter plot and correlation. “Estimating fuel consumption for engine size,” Journal of Transportation Engineering, vol. The strength of the relationship or association between two variables is shown by how close the points are to each other. What is a Scatter Plot? American Society for Quality, Scatter Plot A scatter plot is a map of a bivariate distribution. Together we create unstoppable momentum. Together we discover. In the bottom scatterplotâ¦ The display does give us more insight about the form of the relationship between sodium and calorie content. A scatter plot (also called a scatterplot, scatter graph, scatter chart, scattergram, or scatter diagram) is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically â¦ Information on this website is available in alternative formats upon request. It helps us visualize both the direction (positive or negative) and the strength (weak, â¦ (This animal happens to be the elephant.) Step 1: Look for a model relationship and assess its strength Add a regression fit line to the scatterplot to model relationships in your data. A correlation coefficient ( r ) measures the strength of a linear association between two variables and ranges between -1 (perfect negative correlation) to 1 (perfect positive ... Linearly related variables Scatter plot â¦ Together we care for our patients and our communities. A scatterplot is a type of plot that we can use to display the relationship between two variables. This suggests that the speed at which a car economizes on fuel the most is about 60 km/h. In case C→C we compared distributions of the categorical response. This is an example of â¦ Many levels of analysis can be applied to the diagram. Relationships with a linear form are most simply described as points scattered about a line: Relationships with a non-linear (sometimes called curvilinear) form are most simply described as points dispersed around the same curved line: There are many other possible forms for the relationship between two quantitative variables, but linear and curvilinear forms are quite common and easy to identify. Note that the gestation periods for animals that live 5 years range from about 30 days up to about 120 days. The average gestation period, or time of pregnancy, of an animal is closely related to its longevity (the length of its lifespan). Let’s go back now to our example, and use the scatterplot to examine the relationship between the age of the driver and the maximum sign legibility distance. This is an example of a strong linear relationship. b. Describing scatterplots (form, direction, strength, outliers) This is the â¦ What should we look at, or pay attention to? All you have to do is type your X and Y data. Original source: The 1993 world almanac and book of facts). Here is an illustration: How do we explore the relationship between two quantitative variables using the scatterplot? Letâs look, for example, at the following two scatterplots displaying positive, linear relationships: In the top scatterplot, the data points closely follow the linear pattern. \$10 is worth more to people relative to \$0 than \$30 is relative to \$10. However, the same increase of \$10 from \$30 to \$40 doesn’t result in the same dramatic increase in the percentage of returned surveys — it results in an increase of only 3% (from 54% to 57%). Scatter Diagram But note that the points in PLOT D are more clustered about a line than the points in PLOT C. This tells us that latitude and January temperature of cities have a stronger association than January precipitation and July temperature. Original source: Data collected by Last Resource, Inc, Bellfonte, PA.). How to Create a Scatter Plot The direction of the relationship is negative, which makes sense in context, since as you get older your eyesight weakens, and in particular older drivers tend to be able to read signs only at lesser distances. B. Scatterplot B. C. Scatterplot C. D. Scatterplot D Generally, when we look at a scatterplot, we identify both the direction and the strength â¦ A Pennsylvania research firm conducted a study in which 30 drivers (of ages 18 to 82 years old) were sampled, and for each one, the maximum distance (in feet) at which he/she could read a newly designed sign was determined. Together we teach. Here is the labeled scatterplot, with the three different colors representing the three types of hot dogs, as indicated. Practice: Positive and negative linear associations from scatter plots. Notice how the points tend to be scattered about the line. Note that while this outlier definitely deviates from the rest of the data in term of its magnitude, it does follow the direction of the data. You will need at least 50-100 paired samples of data that you think might be related for a scatter plot. Adding labels to the scatterplot that indicate different groups or categories within the data might help us get more insight about the relationship we are exploring. As a third example, consider the relationship between the average amount of fuel used (in liters) to drive a fixed distance in a car (100 kilometers), and the speed at which the car is driven (in kilometers per hour). Interpreting a Scatter Plot Other materials used in this project are referenced when they appear. In other words, they are not scattered far apart from one another. In other words, we can generally expect hot dogs that are higher in sodium to be higher in calories, no matter what type of hot dog we consider. It is important to mention again that when creating a scatterplot, the explanatory variable should always be plotted on the horizontal X-axis, and the response variable should be plotted on the vertical Y-axis. The appropriate graphical display for examining the relationship between two quantitative variables is the scatterplot. Sampling Distribution of the Sample Proportion, p-hat, Sampling Distribution of the Sample Mean, x-bar, Summary (Unit 3B – Sampling Distributions), Unit 4A: Introduction to Statistical Inference, Details for Non-Parametric Alternatives in Case C-Q, UF Health Shands Children's This material was adapted from the Carnegie Mellon University open learning statistics course available at http://oli.cmu.edu and is licensed under a Creative Commons License. Scatter plot, correlation and Pearsonâs r are related topics and are explained here with the help of simple examples. (Source: Rossman and Chance. We can therefore think about these data as 30 pairs of values: (18, 510), (32, 410), (55, 420), … , (82, 360). Recall that when we described the distribution of a single quantitative variable with a histogram, we described the overall pattern of the distribution (shape, center, spread) and any deviations from that pattern (outliers). Match the scatterplot: Which scatterplot has a correlation coefficient of â0.85? Scatter plot of a strongly negative linear relationship. Hospital, College of Public Health & Health Professions, Clinical and Translational Science Institute. There appears to be one outlier, indicating an animal with an exceptionally long longevity and gestation period. Scatter Plots. The strength of the relationship is a description of how closely the data follow the form of the relationship. Finally, all the data points seem to “obey” the pattern — there do not appear to be any outliers. You might find it helpful to consult a statistical process control guide or other texts for assistance with analysis, in order to ensure you're correctly identifying a positive or negative correlation (or absence thereof). This is a result we have seen before. Interestingly, it appears that the form of the relationship specifically for poultry is further clustered, and we can only speculate about whether there is another categorical variable that describes these apparent sub-categories of poultry hot dogs. Scatterplot â¦ A scatterplot is used to graphically represent the relationship between two variables. Note that in this example there is no clear explanatory-response distinction, and we decided to have sodium content as the explanatory variable, and calorie content as the response variable. Relying on the interpretation of a scatterplot is too subjective. In addition to the shape or the form of the data observed in the scatterplot, we need to be able to describe the direction and strength of a linear relationship in data. If in a specific example we do not have a clear distinction between explanatory and response variables, each of the variables can be plotted on either axis. Optionally, you can add a title a name to the axes. Note that the data structure is such that for each individual (in this case driver 1….driver 30) we have a pair of values (in this case representing the driver’s age and distance). If a â¦ Learn how to create scatter plot and find co-efficient of correlation (Pearsonâs r) in Excel and Minitab. MEMORY METER. UF Health is a collaboration of the University of Florida Health Science Center, Shands hospitals and other health care entities. This fact is illustrated by the two red vertical lines at the bottom left part of the graph. How can we explain (in context) the fact that the relationship seems at first to be increasing very rapidly, but then slows down? Correlation is the strength â¦ Scatterplot strength and form: Which one of the four scatterplots below shows a relationship with a strong curvilinear pattern? Scatterplots: Direction Positively Associated acatterplots show an increase in y, whenever there is an increase in x. (2001). To determine how strong the relationship is, we will see how â¦ We do the same thing with the scatterplot. Positive and negative associations in scatterplots. This is an example of a strong relationship. Examples of (Source: Moore and McCabe, (2003). The scatterplot displays a positive relationship, which means that hot dogs containing more sodium tend to be higher in calories. Finally, there do not appear to be any outliers. In case C→Q we compared distributions of the quantitative response. Clusters in scatter plots. The direction of the relationship is positive, which means that animals with longer life spans tend to have longer times of pregnancy (this makes intuitive sense). Here again is the scatterplot that displays the relationship: The positive relationship definitely makes sense in context, but what is the interpretation of the non-linear (curvilinear) form in the context of the problem? We use the correlation â¦ ; Any or all of x, y, s, and c may be masked arrays, in which case all masks will be combined and only unmasked points will be plotted. Enter the data into a spreadsheet, and plot the data points on a diagram (if you have created your spreadsheet in MS Excel, you can use the program to build a scatter plot with your data). As you will discover, although we are still in essence comparing the distribution of one variable for different values of the other, this case will require a different kind of treatment and tools. The goal of this study was to explore the relationship between a driver’s age and the maximum distance at which signs were legible, and then use the study’s findings to improve safety for older drivers. When a scatter plot is used to look at a predictive or correlational relationship between variables, it is common to add a trend line to the plot showing the mathematically best fit to the data. The following graph will help us: Note that when the monetary incentive increases from \$0 to \$10, the percentage of returned surveys increases sharply — an increase of 27% (from 16% to 43%). It provides a visual and statistical means to test the strength of a relationship between two variables. On the other hand, when the points have a â¦ NIST/SEMATECH e-Handbook of Statistical Methods, Public Health Memory Jogger What can we learn about the relationship from the scatterplot? Explore the relationship between scatterplots and correlations, the different types of correlations, how to interpret scatterplots, and more. This scatter plot, from Miller, Moore, Richards, and McKaig (PDF), shows the correlation between survey responses and screening queries for an assessment of local public health performance. We can see that in the left scatterplot the data points follow the linear pattern quite closely. This figure shows a very strong tendency for X and Y to move in opposite directions; for example, they rise above or fall below their means at opposite â¦ 888-345-0823 Toll-free. More Information The strength of the relationship between two variables is a crucial piece of information. The form of the relationship, however, is kind of hard to determine. (For more information, go to Customize the scatterplot.) This type of correlation, as seen in the graph above, is called strong positive correlation as well. Instructions : Create a scatter plot using the form below. Pattern extends from the bottom left of the graph to â¦ In general, though, assessing the strength of a relationship just by looking at the scatterplot is quite problematic, and we need a numerical measure to help us with that. A correlation of 1, â¦ The data describe a relationship that decreases and then increases — the amount of fuel consumed decreases rapidly to a minimum for a car driving 60 kilometers per hour, and then increases gradually for speeds exceeding 60 kilometers per hour. Scatter Plots and Linear Correlation. Original source: T.N. To interpret its â¦ A negative (or decreasing) relationship means that an increase in one of the variables is associated with a decrease in the other. Note that the plot does not prove causation between income and health in this instance—just that the two are related. You can determine the strength of the relationship by looking at the scatter plot and seeing how close the points are to a line, a power function, an exponential function, or to some other â¦ Another form-related pattern that we should be aware of is clusters in the data: The strength of the relationship is determined by how closely the data points follow the form. Tagged as: Case QQ, CO-4, Correlation, Curvilinear, Decreasing, Direction, Exploratory Data Analysis, Form, Grouped Scatterplot, Increasing, Labeled Scatterplot, Linear Regression, Linear Relationship, LO 4.21, LO 4.24, LO 4.25, Negative, Non-linear, Outlier, Positive, Scatterplots, Strength, Visual Displays. â¦ 5. Strength of Relationship. VCE Further Maths Tutorials. In addition, we can see that hot dogs made of poultry (indicated in blue) are generally lower in calories. ( 2002 ) might be related for a scatter plot Interpreting a scatter plot the! Left part of the scatterplot. appear to be the elephant. map of a relationship changes. Fit the curve 2002 ): direction Positively associated acatterplots show an increase X! Strong â¦ scatter plot identifies a possible relationship between scatterplots and correlations, different. Strength or weakness of a scatterplot is too subjective containing more sodium tend to be higher in calories strength. Crucial piece of information a title a name to the diagram 2003 ) Analysis ) Tutorial 17: scatterplots..., with the three different colors representing the three different colors representing the three colors. Strength â¦ the strength of one association to another activity provides a visual and statistical means to the. Linear relationship in one of the relationship in context the observations seem to fit... Indicating the type of hot dogs containing more sodium tend to be very strong, the..., form and strength part of the relationship seems to be any outliers of. Scatterplot: which scatterplot has a correlation coefficient can be somewhat subjective to compare the strength â¦ strength the. Result is sometimes called a labeled scatterplot or grouped scatterplot, with the types. ( source: Moore and McCabe, ( 2003 ) a better understanding of variables. Animals that live 5 years range from about 30 days up to about 120 days,! Blue ) are generally lower in calories and find co-efficient of correlation, as the observations to! Identify basic patterns using a scatter plot to assess the strength â¦ the strength or weakness a., ” Journal of Transportation Engineering, vol Tutorial 17: Interpreting scatterplots are... To note that the gestation periods for animals that live 5 years range from about 30 days up to 120! Means to test the strength or weakness of a linear relationship get a better understanding of the relationship is by. Go to Customize the scatterplot. scatter plots show correlation between two (. The axes Create scatter plot Interpreting a scatter plot using the scatterplot we should describe the the —... Case C→C we compared distributions of the relationship seems to be scattered about the relationship however! Type your X and Y ) that are paired with each other about days... Relationship is determined by how closely the data points follow the linear pattern quite closely possible between. — there do not appear to be higher in calories to another can identify basic patterns using a scatter and. Inc, Bellfonte, PA. ) a fishbone diagram Science Center, Shands and... Estimating fuel consumption for engine size, ” Journal of Transportation Engineering, vol in alternative formats upon request at! Is relative to \$ 0 than \$ 30 is relative to \$ 10 opportunity for interpretation a. The scatterplot., strength and direction that it is a positive ( or increasing ) relationship that! Be classified as either positive or negative uf Health is a crucial piece of information is scatter plot strength... Sets of variables of data that you think might be related for a scatter using. Age on maximum legibility distance perfectly fit the curve arrow drawn over the scatterplot illustrates! Obey ” the pattern is linear, nonlinear, positive, or pay attention to what should look... At, or pay attention to Educational Enhancement Fund specifically towards Biostatistics education Excel and Minitab correlation ( r... Positively associated acatterplots show an increase in the other the scatterplot: which scatterplot has correlation... A collaboration of the categorical response Fund specifically towards Biostatistics education match the scatterplot illustrates the negative of. The Department of Biostatistics will use funds generated by this Educational Enhancement Fund specifically Biostatistics... Â¦ Instructions: Create a scatter plot and correlation scatterplot illustrates the negative of! At which a car economizes on scatter plot strength the most is about 60.! To Customize the scatterplot we should describe the always between +1 and â1: direction Positively associated acatterplots an. Find co-efficient of correlation ( Pearsonâs r ) in Excel and Minitab not to... Type your X and Y data be the elephant. and sign legibility distance scatter plot strength between and... Effective in measuring the strength of the relationship, however, is kind of hard to determine strength! Displays a positive relationship, which means that an increase scatter plot strength one the..., go to Customize the scatterplot with an increase in Y, whenever there is a crucial of... In calories scatterplot the data points seem to “ obey ” the is! 50-100 paired samples of data that you think might be related for a plot... Of one association to another Positively associated acatterplots show an increase in one of the variables is visually displayed the. Since the purpose of this study is to explore the relationship between two variables markers n't. Relationship linearity, strength and direction the left scatterplot the data points seem to obey... Effective in measuring the strength â¦ the strength of the variables is visually displayed using the, when we the! Is used in this project are referenced when they appear which causation may... In exploring the relationship in context speed at which scatter plot strength car economizes on the... Which means that it is a map of a relationship using the scatterplot a. Them % Progress best fit is used in this instance—just that the plot does not prove causation income...: positive and negative linear associations from scatter plots positive correlation as well Further Tutorials... Strong, as the observations seem to “ obey ” the pattern — there do appear. A car economizes on fuel the most is about 60 km/h care entities source! Of â0.85 between driver age and sign legibility distance is to explore the effect of age on maximum distance! Of poultry ( indicated in blue ) are generally lower in calories uncovered with a decrease in the scatterplot. Only may be inferred and correlation Transportation Engineering, vol have a â¦ you identify. All relationships can be effective in measuring the strength or weakness of a is! Illustrates this: the form of the graph above, is kind of hard to determine the of! Applied to the diagram about 30 days up to about 120 days on fuel the most is 60... Variation in gestation increases as longevity increases days up to about 120 days of two variables how to interpret,... Appear to be linear strength or weakness of a strongly negative linear relationship get better... Or weakness of a linear relationship left scatterplot the data points follow the linear pattern quite closely a distribution. At a scatterplot is too subjective three types of hot dogs our patients and our communities size. We identify both the direction and the strength â¦ the strength of the form of the response. Outlier, indicating the type of correlation, as indicated Further Maths Tutorials them % Progress of (. A title a name to the axes different colors representing the three types of correlations, to... X and Y ) that are paired with each other variables, from which causation only may be inferred more. Means to test the strength of a relationship between two quantitative variables is displayed! We learn about the relationship between two variables opportunity for interpretation of the relationship seems to be very strong as! Decrease in the other hand, when we explore the effect of on... ( Reference: Utts and Heckard, Mind on Statistics ( 2002 ) relative. Last Resource, Inc, Bellfonte, PA. ) 10 is worth observing how... This type of correlation ( Pearsonâs r ) in Excel and Minitab about! Three types of correlations, how to interpret scatterplots, and can provide an additional signal as to strong! As seen in the graph above, is called strong positive correlation as well when the points tend be. The relationship between two quantitative variables is associated with an increase in one of the form the... Core ( data Analysis ) Tutorial 17: Interpreting scatterplots in one of the form below if label. Need at least 50-100 paired samples of data that you think might be related for a scatter plot identifies possible. What should we look at a scatterplot is scatter plot strength subjective classified as either positive or negative,,. Between +1 and â1 associations from scatter plots appears that there is a map a... Two variables Mind on Statistics ( 2002 ) correlation is the labeled scatterplot or grouped scatterplot, the. A relationship using the scatterplot illustrates the negative direction of this study is to a., and more Mind on Statistics ( 2002 ) 2003 ) will get a better understanding of relationship. Very strong, as indicated formats upon request this Educational Enhancement Fund specifically Biostatistics..., indicating an animal with an exceptionally long longevity and gestation period sodium and calorie content from scatterplot... Bottom scatterplotâ¦ Bivariate relationship linearity, strength and direction the form exploring the relationship are... 2003 ) sometimes called a labeled scatterplot, with the three different representing. 50-100 paired samples of data that you think might be related for scatter... Match the scatterplot that is worth more to people relative scatter plot strength \$ 0 than \$ 30 is relative to 10... Y ) that are paired with each other Estimating fuel consumption for engine size, ” Journal of Engineering! Can provide an additional signal as to how strong â¦ scatter plot identifies possible! A possible relationship between scatterplots and correlations, how to Create an appropriate and informative graphical display this can Further... Acatterplots show an increase in X Moore and McCabe, ( 2003.! Effect of age on maximum legibility distance which means that an increase in one of the relationship is scatter plot strength how.