Why is the mean(p^) = mean(X)/n ? Well, it's not, as you said earlier mean(p^)= p. I'll show you why, but forget proportions for a minute. When you draw a sample, you get a random variable with a particular distribution. Six-sided dice have a discrete uniform distribution on values, so rolling a die gives you a variable with mean 7/2 and variance 35/12. Drawing a Normal(0,1) variable gives you a variable with mean 0 and variance 1. We often use the term expected value or expectation for means (partially just to avoid using the word "mean" a billion times when talking about sums and their means). Expectation has a nice property called linearity . If you have one variable X and a constant k, E[kx] = k E[X]. If you have a bunch of variables X_1,X_2,...,X_n, then E[X_1 + X_2 + ... + X_n] = E[X_1] + E[X_2] + ... + E[X_n]. If those variables all have the same expected value E[X], then the expectation of the sum is E[X_1 + X_2 + ... + X_n] = n E[X]. The expectation of the mean of the variables X_1,X_2,...,X_n is E[1/n * (X_1 + X_2 + ... + X_n)] = 1/n E[X_1 + X_2 + ... + X_n] = 1/n * n E[X] = E[X]. Okay, back to proportions. The key to proportions is remembering that a proportion is a mean (you sum values and divide by the number of values). The sample proportion p^ = 1/n (X_1 + X_2 + ... + X_n). That's also the sample mean. In proportions, each of your X_i are either 0 (with probability 1-p) or 1 (with probability p) and E[X_i] = p. Thus, the expected value of the sum of samples is np, and the expected mean of the samples (p^) is 1/n *np = p. So, the sample mean is p, and since sample means are sample proportions, E[p^] = p. Edit for bad formatting. Answer from n_eff on reddit.com
Omni Calculator
omnicalculator.com › statistics › p-hat
P-Hat Calculator
January 18, 2024 - Use this p-hat calculator to determine the sample proportion according to the number of occurrences of an event and the sample size.
Statistics LibreTexts
stats.libretexts.org › bookshelves › applied statistics › biostatistics - open learning textbook › unit 3b: sampling distribution
Sampling Distribution of the Sample Proportion, p-hat - Statistics LibreTexts
September 27, 2024 - If repeated random samples of a given size n are taken from a population of values for a categorical variable, where the proportion in the category of interest is p, then the mean of all sample proportions (p-hat) is the population proportion (p).
How to Calculate PHat?
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Sample distribution of sample proportions mean formula explanation
Why is the mean(p^) = mean(X)/n ? Well, it's not, as you said earlier mean(p^)= p. I'll show you why, but forget proportions for a minute. When you draw a sample, you get a random variable with a particular distribution. Six-sided dice have a discrete uniform distribution on values, so rolling a die gives you a variable with mean 7/2 and variance 35/12. Drawing a Normal(0,1) variable gives you a variable with mean 0 and variance 1. We often use the term expected value or expectation for means (partially just to avoid using the word "mean" a billion times when talking about sums and their means). Expectation has a nice property called linearity . If you have one variable X and a constant k, E[kx] = k E[X]. If you have a bunch of variables X_1,X_2,...,X_n, then E[X_1 + X_2 + ... + X_n] = E[X_1] + E[X_2] + ... + E[X_n]. If those variables all have the same expected value E[X], then the expectation of the sum is E[X_1 + X_2 + ... + X_n] = n E[X]. The expectation of the mean of the variables X_1,X_2,...,X_n is E[1/n * (X_1 + X_2 + ... + X_n)] = 1/n E[X_1 + X_2 + ... + X_n] = 1/n * n E[X] = E[X]. Okay, back to proportions. The key to proportions is remembering that a proportion is a mean (you sum values and divide by the number of values). The sample proportion p^ = 1/n (X_1 + X_2 + ... + X_n). That's also the sample mean. In proportions, each of your X_i are either 0 (with probability 1-p) or 1 (with probability p) and E[X_i] = p. Thus, the expected value of the sum of samples is np, and the expected mean of the samples (p^) is 1/n *np = p. So, the sample mean is p, and since sample means are sample proportions, E[p^] = p. Edit for bad formatting. More on reddit.com
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October 4, 2021mean - Find the variance of p-hat - Cross Validated
We have not discussed $\hat p$ in my probability and statistics course and a problem involving it is on our hw this week after learning about discrete distributions. The problem states "Let the ran... More on stats.stackexchange.com
When do you use P(hat) combined for checking the normal condition of a two sample z- test?
When they’re asking if there is a difference between the two proportions (when p1 doesn’t equal p2) for the alternative hypothesis More on reddit.com
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April 16, 2022How do I find p-hat?
To find p-hat (i.e., sample proportion), you need to follow the next steps:
- Take the number of occurrences of an event or the number of successful outcomes.
- Divide it by the sample size.
- That's all! You have calculated p-hat.
omnicalculator.com
omnicalculator.com › statistics › p-hat
P-Hat Calculator
What is the meaning of p-hat?
P-hat coveys the sample proportion, the ratio of certain events or characteristics occurring in a sample to the sample size. It can equal or differ from population proportion, which conveys a proportion of a particular feature associated with a population.
omnicalculator.com
omnicalculator.com › statistics › p-hat
P-Hat Calculator
What does it mean if p-hat equals 0.6 in a political poll?
If p-hat equals 0.6 in a political poll, 60% of voters from the sample support a particular event or a candidate. P-hat is the ratio of the number of occurrences of a particular event to the sample size and is often reported as a percentage in polls.
omnicalculator.com
omnicalculator.com › statistics › p-hat
P-Hat Calculator
Videos
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The Sampling Distribution of P-hat, The Sample Proportion. Includes ...
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Sampling Distribution of p hat - YouTube
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How To Find P Hat Statistics? - The Friendly Statistician - YouTube
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Sampling Distribution of the Sample Proportion (7.4) - YouTube
7.2 Sampling Distribution of p-hat - YouTube
Wumbo
wumbo.net › symbols › p-hat
P Hat Symbol (p̂)
In statistics, the p-hat symbol (written as p̂, with a “hat” or “caret” over the letter p) is used to represent the proportion of a sample with a particular characteristic or outcome.
GeeksforGeeks
geeksforgeeks.org › mathematics › how-to-calculate-p-hat
How to Calculate P-Hat? - GeeksforGeeks
February 15, 2024 - In summary, P-hat (p̂) is calculated by dividing the number of successes by the total number of observations in the sample.
Wikipedia
en.wikipedia.org › wiki › Population_proportion
Population proportion - Wikipedia
October 16, 2025 - {\displaystyle -z^{*}<{\frac {{\hat {p}}-P}{\sqrt {\frac {{\hat {p}}(1-{\hat {p}})}{n}}}}<z^{*}\Rightarrow -z^{*}{\sqrt {\frac {{\hat {p}}(1-{\hat {p}})}{n}}}<{\hat {p}}-P<z^{*}{\sqrt {\frac {{\hat {p}}(1-{\hat {p}})}{n}}}\Rightarrow -{\hat {p}}-z^{*}{\sqrt {\frac {{\hat {p}}(1-{\hat {p}})}{n}}}<-P<-{\hat {p}}+z^{*}{\sqrt {\frac {{\hat {p}}(1-{\hat {p}})}{n}}}\Rightarrow {\hat {p}}-z^{*}{\sqrt {\frac {{\hat {p}}(1-{\hat {p}})}{n}}}<P<{\hat {p}}+z^{*}{\sqrt {\frac {{\hat {p}}(1-{\hat {p}})}{n}}}} From the algebraic work done above, it is evident from a level of certainty ... In general the formula used for estimating a population proportion requires substitutions of known numerical values.
Reddit
reddit.com › r/askstatistics › sample distribution of sample proportions mean formula explanation
r/AskStatistics on Reddit: Sample distribution of sample proportions mean formula explanation
October 4, 2021 -
Disclaimer: I'm learning medical statistics (I'm a doctor) and I'm no expert in math so that may be a dumb question to you all.
I'm learning the sampling distribution in general and now I got to the sampling distribution of sample proportions, which in theory it makes sense but I can't get the mathematical explanation of it.
Let me explain:
I know that the p-hat (I'll write it like p^) is referred to the sample proportion and the formula is
p^= X/n
and the mean of the sample 's formula is
mean(X) = np
Where:
X is the number of "successes" ( like people who voted me) in my sample
n is the total number of people in my sample (people who voted me and people who didn't)
Now let's suppose that the entire population has a proportion= p
When looking for the sample distribution of the sample proportions they all say, without that many mathematical explanations, that the mean(p^)= p. This makes sense theoretically but not mathematically, to me.
When trying to explain it mathematically (very very few articles on the web) they all say:
mean(p^)= mean(X)/n -> np/n -> p
My question
Why is the mean(p^) = mean(X)/n ?
Isn't "X" referred to the successes in one sample? Why do I take the mean of ONE sample and divide it for "n"?
When reading "np" -> isn't "n" the number of people in the SAMPLE? And isn't the "n" in the denominator the number of samples? Why do they cancel each other since they have different meaning?
This is terribly confusing for people like me and basically none in the web tries to even explain that formula, they all just say mean(p^) = p and that's it.
Top answer 1 of 2
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Why is the mean(p^) = mean(X)/n ? Well, it's not, as you said earlier mean(p^)= p. I'll show you why, but forget proportions for a minute. When you draw a sample, you get a random variable with a particular distribution. Six-sided dice have a discrete uniform distribution on values, so rolling a die gives you a variable with mean 7/2 and variance 35/12. Drawing a Normal(0,1) variable gives you a variable with mean 0 and variance 1. We often use the term expected value or expectation for means (partially just to avoid using the word "mean" a billion times when talking about sums and their means). Expectation has a nice property called linearity . If you have one variable X and a constant k, E[kx] = k E[X]. If you have a bunch of variables X_1,X_2,...,X_n, then E[X_1 + X_2 + ... + X_n] = E[X_1] + E[X_2] + ... + E[X_n]. If those variables all have the same expected value E[X], then the expectation of the sum is E[X_1 + X_2 + ... + X_n] = n E[X]. The expectation of the mean of the variables X_1,X_2,...,X_n is E[1/n * (X_1 + X_2 + ... + X_n)] = 1/n E[X_1 + X_2 + ... + X_n] = 1/n * n E[X] = E[X]. Okay, back to proportions. The key to proportions is remembering that a proportion is a mean (you sum values and divide by the number of values). The sample proportion p^ = 1/n (X_1 + X_2 + ... + X_n). That's also the sample mean. In proportions, each of your X_i are either 0 (with probability 1-p) or 1 (with probability p) and E[X_i] = p. Thus, the expected value of the sum of samples is np, and the expected mean of the samples (p^) is 1/n *np = p. So, the sample mean is p, and since sample means are sample proportions, E[p^] = p. Edit for bad formatting.
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Note that you're dealing with a binomial model here https://en.wikipedia.org/wiki/Binomial_distribution The problem you have is that you're flipping between two different meanings for X. This is a common problem when you (or your sources) don't all have the same definition for X in every place. You have to translate everything to a single definition -- you can't just conflate formulas across different meanings for the same symbol. If X is the number of successes, per your post (under the model X itself is then binomial(n,p)): p-hat = X/n E(X/n) = E(X)/n = np/n = p (see the above link). If you have Y being the sample proportion, Y = X/n, then p-hat = Y and E(Y) = p Either way, E(p-hat) = p
Richland College
people.richland.edu › james › lecture › m170 › ch08-pro.html
Stats: Estimating the Proportion
The maximum error of the estimate is given by the formula for E shown. The Z here is the z-score obtained from the normal table, or the bottom of the t-table as explained in the introduction to estimation. The z-score is a factor of the level of confidence, so you may get in the habit of writing it next to the level of confidence. When you're computing E, I suggest that you find the sample proportion, p hat...
Sciencing
sciencing.com › calculate-phat-8384855
How To Calculate P-Hat - Sciencing
March 24, 2022 - The actual calculation of p-hat is not challenging. To do it, you need two numbers. One is the sample size (n) and the other is the number of occurrences of the event or parameter in question (X). The equation for p-hat is p-hat = X/n.
Fiveable
fiveable.me › all key terms › ap statistics › p-hat (sample proportion)
P-hat (sample proportion) Definition - AP Statistics Key Term | Fiveable
p-hat is a statistic that represents the proportion of a certain outcome in a sample, calculated as the number of successes divided by the total number of observations in that sample. This concept is crucial when estimating population proportions, especially in constructing confidence intervals, ...
University of North Dakota
cs.uni.edu › ~campbell › stat › inf6.html
Proportions
However. sometimes it is convenient to use proportions (e.g., the fraction of the population who approve of Clinton) rather than the actual count (the number of people who approve of Clinton). If the sample size is n, the proportion can be obtained from the count by division by n: p-hat = X/n ...
Elgin
faculty.elgin.edu › dkernler › statistics › ch08 › 8-2.html
Chapter 8
In general, if we let x = the number with the specific characteristic, then the sample proportion, , (read "p-hat") is given by:
Pearson
pearson.com › channels › statistics › exam-prep › set › default › statistics-final › what-is-the-formula-for-calculating-the-sample-proportion-p-hat
Statistics Final | Practice Questions & Video Solutions
p hat = n / x · B · p hat = (x + n) / 2 · C · p hat = x * n · D · p hat = x / n · AI tutor0 · Show Answer ·
Stats4stem
stats4stem.org › r-one-proportion
STATS4STEM
> p.hat = 47/64 # calculate sample proportion > p.hat [1] 0.734375 > > p.null = .60 # define null proportion > n = 64 # define sample size > > # Check conditions for inference > n*p.null [1] 38.4 > n*(1 - p.null) [1] 25.6 · Conclusion: n*phat and n*qhat are both greater than 10. Therefore, we can proceed with our inference calculations. 3. Next, we will calculate our test statistic for this problem using the following formula.
Academic Help
academichelp.net › stem › statistics › what-is-p-hat.html
What is P-hat in Statistics: Understanding Sampling and Estimation
August 10, 2023 - To calculate p-hat, two pieces of information are required: the sample size (n) and the number of occurrences of the event or parameter of interest (X) within the sample. The formula for p-hat is simple: p̂ = X/n.
Penn State University
online.stat.psu.edu › stat200 › lesson › 2 › 2.1 › 2.1.1
2.1.1 - One Categorical Variable | STAT 200
The symbol for a sample proportion is \(\widehat{p}\) and is read as "p-hat." The symbol for a population proportion is \(p\). The formula for a sample proportion may also be written as \(\widehat p = \frac{x}{n}\) where \(x\) is the number in the sample with the trait of interest and \(n\) ...
Sjsu
www2.sjsu.edu › faculty › gerstman › StatPrimer › conf-prop.htm
Inference for a Proportion
s2 = npq (where q = 1 - p). Concurrently, when the normal approximation holds, sample proportion p^ is normally distributed with mean p and variance pq/n.
Study.com
study.com › math courses › statistics 101: principles of statistics
Finding Confidence Intervals for Proportions: Formula & Example - Lesson | Study.com
March 8, 2016 - Again, all we have to do is plug and chug into the formulas I gave you before to find our confidence interval. p-hat (+/-) z x s_p-hat = 0.6 (+/-) 2.58(0.015492) = 0.56 to 0.64 = 56%-64%.
Statistics LibreTexts
stats.libretexts.org › under construction › introductory statistics with google sheets (kesler) › 7: confidence intervals
7.3: A Population Proportion - Statistics LibreTexts
March 29, 2022 - If X is a binomial random variable, then X ~ B(n, p) where n is the number of trials and p is the probability of a success. To form a proportion, take X, the random variable for the number of successes and divide it by n, the number of trials ...