https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:prob-comb/x9e81a4f98389efdf:expected-value/v/expected-value-of-a-discrete-random-variable 2022-10-07T00:22:18.752150461Z Mean (expected value) of a discrete random variable We can calculate the mean (or expected value) of a discrete random variable as the weighted average of all the outcomes of that random variable based on their probabilities. We interpret expected value as the predicted average outcome if we looked at that random variable over an infinite number of trials. video https://cdn.kastatic.org/ka_thumbnails_cache/74a42b52-15cb-443c-bd3d-cd165775c4c0_1280_720_base.png Mean (expected value) of a discrete random variable We can calculate the mean (or expected value) of a discrete random variable as the weighted average of all the outcomes of that random variable based on their probabilities. We interpret expected value as the predicted average outcome if we looked at that random variable over an infinite number of trials. https://cdn.kastatic.org/ka-youtube-converted/qafPcWNUiM8.mp4/qafPcWNUiM8.mp4 272 Random variables and sampling https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/a/expected-value-basic 2021-04-29T07:58:49Z Expected value (basic) article https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/e/expected_value 2026-03-24T23:41:07.080866137Z Expected value Compute the expected value given a set of outcomes, probabilities, and payoffs Mark Wittels exercise https://www.khanacademy.org/math/ap-statistics/random-variables-ap/xfb5d8e68:mean-standard-deviation-random-variables/v/variance-and-standard-deviation-of-a-discrete-random-variable 2024-01-19T18:34:27.70568811Z Variance and standard deviation of a discrete random variable We learn how to calculate the mean and standard deviation of a discrete random variable. The concept of a random variable is explained, along with methods to calculate its expected value (mean) and measure its spread (variance and standard deviation). A practical example makes the concept easier to understand. video https://cdn.kastatic.org/ka_thumbnails_cache/1480a2e2-c783-4cc6-b18d-f26efd6e3e77_1280_720_base.png Variance and standard deviation of a discrete random variable We learn how to calculate the mean and standard deviation of a discrete random variable. The concept of a random variable is explained, along with methods to calculate its expected value (mean) and measure its spread (variance and standard deviation). A practical example makes the concept easier to understand. https://cdn.kastatic.org/ka-youtube-converted/2egl_5c8i-g.mp4/2egl_5c8i-g.mp4 386 Random variables and sampling https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/v/random-variables 2020-07-26T09:23:46Z Random variables Basic idea and definitions of random variables Sal Khan video https://cdn.kastatic.org/googleusercontent/qvRdke9M6v3Qpb1vuYZsEoZeINwGrsvcCs5yqZEV5wjs2Tq6BO4aQY1pMvIrKOTHrONu5sjnmgucCBokWEAt1MJ_ Random variables Basic idea and definitions of random variables https://cdn.kastatic.org/ka-youtube-converted/3v9w79NhsfI.mp4/3v9w79NhsfI.mp4 332 Random variables and sampling https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/v/discrete-and-continuous-random-variables 2022-11-16T04:22:43.7414897Z Discrete and continuous random variables Discrete random variables can only take on a finite number of values. For example, the outcome of rolling a die is a discrete random variable, as it can only land on one of six possible numbers. Continuous random variables, on the other hand, can take on any value in a given interval. For example, the mass of an animal would be a continuous random variable, as it could theoretically be any non-negative number. Sal Khan video https://cdn.kastatic.org/googleusercontent/C-EUxA7o10ncTqWb2z3f09iyZER2yjTS385EnCOZNmur3MrWqHqiONcNUW0u6_lNg8yGeV8CPihpCspEs_uo-J5z Discrete and continuous random variables Discrete random variables can only take on a finite number of values. For example, the outcome of rolling a die is a discrete random variable, as it can only land on one of six possible numbers. Continuous random variables, on the other hand, can take on any value in a given interval. For example, the mass of an animal would be a continuous random variable, as it could theoretically be any non-negative number. https://cdn.kastatic.org/ka-youtube-converted/dOr0NKyD31Q.mp4/dOr0NKyD31Q.mp4 717 Random variables and sampling https://www.khanacademy.org/math/ap-statistics/random-variables-ap/xfb5d8e68:mean-standard-deviation-random-variables/e/standard-deviation-discrete-random-variable 2021-07-01T02:19:28Z Standard deviation of a discrete random variable Practice calculating the standard deviation of a discrete random variable. Jeff Dodds exercise https://www.khanacademy.org/math/ap-statistics/random-variables-ap/combining-random-variables/a/combining-normal-random-variables 2023-02-22T03:54:13.328982673Z Combining normal random variables article https://www.khanacademy.org/math/ap-statistics/random-variables-ap/combining-random-variables/v/analyzing-distribution-of-sum-of-two-normally-distributed-random-variables 2022-10-07T00:58:19.561984208Z Example: Analyzing distribution of sum of two normally distributed random variables Finding the probability that the total of some random variables exceeds an amount by understanding the distribution of the sum of normally distributed variables. video https://cdn.kastatic.org/ka_thumbnails_cache/d810dd02-8100-45fe-b94b-846071af0e5f_1280_720_base.png Example: Analyzing distribution of sum of two normally distributed random variables Finding the probability that the total of some random variables exceeds an amount by understanding the distribution of the sum of normally distributed variables. https://cdn.kastatic.org/ka-youtube-converted/Z6vANdo1L68.mp4/Z6vANdo1L68.mp4 351 Random variables and sampling https://www.khanacademy.org/math/ap-statistics/density-curves-normal-distribution-ap/normal-distributions-calculations/v/z-table-for-proportion-above 2022-10-07T00:58:19.561984208Z Standard normal table for proportion above Finding the proportion of a normal distribution that is above a value by calculating a z-score and using a z-table. video https://cdn.kastatic.org/ka_thumbnails_cache/7490c1c0-e3e5-4dcc-82a8-2b8393c7f071_1280_720_base.png Standard normal table for proportion above Finding the proportion of a normal distribution that is above a value by calculating a z-score and using a z-table. https://cdn.kastatic.org/ka-youtube-converted/i9FzFfv1rQg.mp4/i9FzFfv1rQg.mp4 307 Random variables and sampling https://www.khanacademy.org/math/ap-statistics/density-curves-normal-distribution-ap/normal-distributions-calculations/v/z-table-for-proportion-below 2022-10-07T00:58:19.561984208Z Standard normal table for proportion below Finding the proportion of a normal distribution that is below a value by calculating a z-score and using a z-table. video https://cdn.kastatic.org/ka_thumbnails_cache/f2012f2f-2bc8-4838-ab75-962bb61b9ec6_1280_720_base.png Standard normal table for proportion below Finding the proportion of a normal distribution that is below a value by calculating a z-score and using a z-table. https://cdn.kastatic.org/ka-youtube-converted/Fo4kitkFB3I.mp4/Fo4kitkFB3I.mp4 260 Random variables and sampling https://www.khanacademy.org/math/ap-statistics/density-curves-normal-distribution-ap/normal-distributions-calculations/v/z-table-for-proportion-between-values 2022-10-07T00:58:19.561984208Z Standard normal table for proportion between values Finding the proportion of a normal distribution that is between two values by calculating z-scores and using a z-table. video https://cdn.kastatic.org/ka_thumbnails_cache/5775492e-4371-49a3-91fc-f1fb66998ed0_1280_720_base.png Standard normal table for proportion between values Finding the proportion of a normal distribution that is between two values by calculating z-scores and using a z-table. https://cdn.kastatic.org/ka-youtube-converted/uwhV0TAPmWc.mp4/uwhV0TAPmWc.mp4 430 Random variables and sampling https://www.khanacademy.org/math/ap-statistics/density-curves-normal-distribution-ap/normal-distributions-calculations/v/finding-z-score-for-a-percentile 2022-11-16T04:22:43.7414897Z Finding z-score for a percentile In any normal distribution, we can find the z-score that corresponds to some percentile rank. If we're given a particular normal distribution with some mean and standard deviation, we can use that z-score to find the actual cutoff for that percentile. In this example, we find what pulse rate represents the top 30% of all pulse rates in a population. video https://cdn.kastatic.org/ka_thumbnails_cache/ac022e1d-8317-48c0-832d-1d0bbc752318_1280_720_base.png Finding z-score for a percentile In any normal distribution, we can find the z-score that corresponds to some percentile rank. If we're given a particular normal distribution with some mean and standard deviation, we can use that z-score to find the actual cutoff for that percentile. In this example, we find what pulse rate represents the top 30% of all pulse rates in a population. https://cdn.kastatic.org/ka-youtube-converted/S5_5KyCVjrU.mp4/S5_5KyCVjrU.mp4 254 Random variables and sampling https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/z-scores/v/ck12-org-normal-distribution-problems-z-score 2020-06-23T07:22:43Z Normal distribution problem: z-scores (from ck12.org) Z-score practice Sal Khan video https://cdn.kastatic.org/googleusercontent/8T6lsuEAcIRtI2uCQe-qbprDFzlZ42YFAl4sWpYxgGFk2Au4yqYL7qxsUDNJ-_lvYezFZHHtd7QGesZpp909xQ Normal distribution problem: z-scores (from ck12.org) Z-score practice https://cdn.kastatic.org/ka-youtube-converted/Wp2nVIzBsE8.mp4/Wp2nVIzBsE8.mp4 468 Random variables and sampling https://www.khanacademy.org/math/ap-statistics/density-curves-normal-distribution-ap/measuring-position/v/comparing-with-z-scores 2022-10-07T00:58:19.561984208Z Comparing with z-scores Use standardized scores—also called z-scores— to compare data points from different distributions. video https://cdn.kastatic.org/ka_thumbnails_cache/02c39d9a-cb03-4ee3-a6e5-795d5fea461f_1280_720_base.png Comparing with z-scores Use standardized scores—also called z-scores— to compare data points from different distributions. https://cdn.kastatic.org/ka-youtube-converted/_rtQGAX5wsQ.mp4/_rtQGAX5wsQ.mp4 210 Random variables and sampling https://www.khanacademy.org/math/ap-statistics/density-curves-normal-distribution-ap/normal-distributions-calculations/v/threshold-for-low-percentile 2022-10-07T00:58:19.561984208Z Threshold for low percentile Find the cutoff for a given lower percentile in a normal distribution video https://cdn.kastatic.org/ka_thumbnails_cache/3ab96c76-65e7-4fe5-803e-f3b8e2e868d1_1280_720_base.png Threshold for low percentile Find the cutoff for a given lower percentile in a normal distribution https://cdn.kastatic.org/ka-youtube-converted/umYWWMfxUCI.mp4/umYWWMfxUCI.mp4 360 Random variables and sampling https://www.khanacademy.org/math/ap-statistics/density-curves-normal-distribution-ap/stats-normal-distributions/v/ck12-org-normal-distribution-problems-qualitative-sense-of-normal-distributions 2021-07-01T17:39:49Z Qualitative sense of normal distributions Discussion of how "normal" a distribution might be. Sal Khan video https://cdn.kastatic.org/googleusercontent/ors031shotyaG_eKmB8Xg1KzOnAyEJC-YH2oJjEIJwU_J8lK33nyoGVkEoUzxdvAtxDLoBKBg6avCpjaNhE0RYDL Qualitative sense of normal distributions Discussion of how "normal" a distribution might be. https://cdn.kastatic.org/ka-youtube-converted/79duxPXpyKQ.mp4/79duxPXpyKQ.mp4 653 Random variables and sampling https://www.khanacademy.org/math/ap-statistics/density-curves-normal-distribution-ap/stats-normal-distributions/v/ck12-org-normal-distribution-problems-empirical-rule 2022-11-16T04:22:43.7414897Z Normal distribution problems: Empirical rule The empirical rule (also called the "68-95-99.7 rule") is a guideline for how data is distributed in a normal distribution. The rule states that (approximately): - 68% of the data points will fall within one standard deviation of the mean. - 95% of the data points will fall within two standard deviations of the mean. - 99.7% of the data points will fall within three standard deviations of the mean. Sal Khan video https://cdn.kastatic.org/googleusercontent/RAuZc0CuZvqkNsXo_OYWJyIThbBI7V6_TBQoIU11Paj9L7K6VxlCLsfOdJZMObG9f6SCopi0AmfM8ov0VN1tWgka Normal distribution problems: Empirical rule The empirical rule (also called the "68-95-99.7 rule") is a guideline for how data is distributed in a normal distribution. The rule states that (approximately): - 68% of the data points will fall within one standard deviation of the mean. - 95% of the data points will fall within two standard deviations of the mean. - 99.7% of the data points will fall within three standard deviations of the mean. https://cdn.kastatic.org/ka-youtube-converted/OhRr26AfFBU.mp4/OhRr26AfFBU.mp4 625 Random variables and sampling https://www.khanacademy.org/math/ap-statistics/density-curves-normal-distribution-ap/stats-normal-distributions/a/basic-normal-calculations 2021-07-01T17:39:48Z Basic normal calculations article https://www.khanacademy.org/math/ap-statistics/density-curves-normal-distribution-ap/normal-distributions-calculations/e/z_scores_2 2024-10-14T00:15:38.200513891Z Normal distribution: Area above or below a point Use a z-table to find the area above or below a given point in some Normal distribution. Sid Patel exercise https://www.khanacademy.org/math/ap-statistics/density-curves-normal-distribution-ap/normal-distributions-calculations/e/z_scores_3 2021-07-01T17:39:47Z Normal distribution: Area between two points Use a z-table to find the area between two given points in some normal distribution. Sid Patel exercise https://www.khanacademy.org/math/ap-statistics/density-curves-normal-distribution-ap/normal-distributions-calculations/e/normal-calculations-3 2024-06-30T20:44:05.927278439Z Normal calculations in reverse Practice finding a value in a normal density curve given an area. Charlotte Auen exercise https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/more-on-normal-distributions/v/normal-distribution-excel-exercise 2020-06-23T07:22:43Z Normal distribution excel exercise (Long-26 minutes) Presentation on spreadsheet to show that the normal distribution approximates the binomial distribution for a large number of trials. Sal Khan video https://cdn.kastatic.org/googleusercontent/su-pKOmscx7Cti5cYuJFvmlr4IqWw5k-k0Wu756aifjyAGw3fYTWOcHwO1wHF_uUDDhZAGu69Dz_KrTSRqgKdEI Normal distribution excel exercise (Long-26 minutes) Presentation on spreadsheet to show that the normal distribution approximates the binomial distribution for a large number of trials. https://cdn.kastatic.org/ka-youtube-converted/yTGEMoaWDCQ.mp4/yTGEMoaWDCQ.mp4 1564 Random variables and sampling https://www.khanacademy.org/math/ap-statistics/sampling-distribution-ap/what-is-sampling-distribution/v/introduction-to-sampling-distributions 2022-10-07T00:58:19.561984208Z Introduction to sampling distributions Introduction to sampling distributions video https://cdn.kastatic.org/googleusercontent/Ru5JWTU2MkRFDMSX6CI6SAYdfdHPAoUlykC4_aHkSzup5QTlj7_BHP57x5nbLw1DwgtaDkJsvjBPgzhUSC-k-UQ7 Introduction to sampling distributions Introduction to sampling distributions https://cdn.kastatic.org/ka-youtube-converted/z0Ry_3_qhDw.mp4/z0Ry_3_qhDw.mp4 438 Random variables and sampling https://www.khanacademy.org/math/ap-statistics/gathering-data-ap/xfb5d8e68:inference-experiments/a/scope-of-inference-random-sampling-assignment 2021-07-01T17:39:50Z Random sampling vs. random assignment (scope of inference) article https://www.khanacademy.org/math/ap-statistics/xfb5d8e68:inference-categorical-proportions/one-sample-z-interval-proportion/v/determining-sample-size-based-on-confidence-and-margin-of-error 2022-10-07T00:58:19.561984208Z Determining sample size based on confidence and margin of error Determining sample size based on confidence level and margin of error video https://cdn.kastatic.org/googleusercontent/gv6WYsLSFop4QPRZBRNWJOwXdoBE8a2XqbfXVUAvmFgmDM5pY52m1WdoBYQTDPocTRh3_OtdtH3PnG0XHEtD1yk Determining sample size based on confidence and margin of error Determining sample size based on confidence level and margin of error https://cdn.kastatic.org/ka-youtube-converted/VyFs7fsWE6w.mp4/VyFs7fsWE6w.mp4 481 Random variables and sampling https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/v/sample-size-for-a-given-margin-of-error-for-a-mean 2022-10-07T00:58:19.561984208Z Sample size for a given margin of error for a mean Calculate the approximate sample size required to obtain a desired margin of error in a confidence interval for a mean. video https://cdn.kastatic.org/googleusercontent/jzKNTSqryEb1eDJfTdJGKD6ZDBTOGTcy5KOqsINkdZtcB0-0LB16s6jIXLE2mSKGR8bIW-Vo_Ws3aIgejrKaXgdV Sample size for a given margin of error for a mean Calculate the approximate sample size required to obtain a desired margin of error in a confidence interval for a mean. https://cdn.kastatic.org/ka-youtube-converted/FSVDJoHZ3rM.mp4/FSVDJoHZ3rM.mp4 462 Random variables and sampling https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/old-confidence-interval-videos/v/small-sample-size-confidence-intervals 2021-04-06T14:32:18Z Small sample size confidence intervals Constructing small sample size confidence intervals using t-distributions Sal Khan video https://cdn.kastatic.org/googleusercontent/uhBXdNgaTWTT_sWwta1ZzIR_pOmamFO645F9aP9r6N79P6HsXb8z_M2J20Z18DSs1DOf9sRCaHchLCMa5fnnDOW- Small sample size confidence intervals Constructing small sample size confidence intervals using t-distributions https://cdn.kastatic.org/ka-youtube-converted/K4KDLWENXm0.mp4/K4KDLWENXm0.mp4 671 Random variables and sampling https://www.khanacademy.org/math/ap-statistics/inference-slope-linear-regression/inference-slope/e/confidence-interval-slope 2023-02-22T03:54:13.328982673Z Confidence interval for slope Practice making a t interval to estimate a population slope from sample data. Jeff Dodds exercise https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-mean/e/sample-size-margin-error-one-sample-t-interval-mean 2024-11-11T02:56:25.272408758Z Sample size and margin of error in a confidence interval for a mean Practice estimating what sample size is required for a given margin of error in a confidence interval for a mean. Jeff Dodds exercise https://www.khanacademy.org/math/ap-statistics/gathering-data-ap/sampling-methods/e/sampling-method-considerations 2021-07-01T17:39:49Z Sampling method considerations Identify the advantages and disadvantages of sampling methods, why we might use one sampling method over another, and how to set up a sample so it represents the population. exercise https://www.khanacademy.org/math/ap-statistics/gathering-data-ap/sampling-methods/v/techniques-for-random-sampling-and-avoiding-bias 2021-07-01T17:39:47Z Techniques for random sampling and avoiding bias Techniques for random sampling and avoiding bias video https://cdn.kastatic.org/googleusercontent/tQ8ZAg4_Int2xx9RIU_am10128q8QetpcPpq-tC5Zs1l7HO_i1EzURjFs_g-yM_BNdsidwaOutO4IHCvNa1x5YU Techniques for random sampling and avoiding bias Techniques for random sampling and avoiding bias https://cdn.kastatic.org/ka-youtube-converted/PdXDLNNXPik.mp4/PdXDLNNXPik.mp4 553 Random variables and sampling https://www.khanacademy.org/math/ap-statistics/gathering-data-ap/sampling-methods/v/techniques-for-generating-a-simple-random-sample 2021-07-01T17:39:46Z Techniques for generating a simple random sample Techniques for generating a simple random sample video https://cdn.kastatic.org/googleusercontent/Jh_CmCnEe7ZNzVHvNbqtDvCQ_1m4HgiBARwDqWLRKisTdEZu5_6XnLvQS4LkaQ-NvEvrlXF5sMcgFwvC0O_Vsb_Z Techniques for generating a simple random sample Techniques for generating a simple random sample https://cdn.kastatic.org/ka-youtube-converted/acfjqWTwee0.mp4/acfjqWTwee0.mp4 386 Random variables and sampling https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-methods-stats/a/sampling-methods-review 2024-02-20T13:25:57.751434196Z Sampling methods review article https://www.khanacademy.org/math/statistics-probability/designing-studies/sampling-methods-stats/a/worksheet-samples-and-surveys 2021-04-29T07:59:30Z Samples and surveys article https://www.khanacademy.org/math/ap-statistics/gathering-data-ap/sampling-methods/e/sampling-methods 2021-07-01T17:39:51Z Sampling methods Practice identifying which sampling method was used in statistical studies, and why it might make sense to use one sampling method over another. Charlotte Auen exercise