3. Kathryn has taught high school or university mathematics for over 10 years. t = x ― − μ 0 s n. May 20, 2024 · A random sample of \(12\) students from a large university yields mean GPA \(2. This calculator finds the probability of obtaining a certain value for a sample mean, based on a population mean, population standard deviation, and sample size. The sampling method is simple random sampling . The mean of the distribution of the sample means is μ¯. 4 days ago · Mean and Standard Deviation Formula. Nov 21, 2023 · The standard deviation of the sampling distribution is called the standard error, and it represents the degree of uncertainty when the population mean is estimated using the sample mean. Use this p-hat calculator to determine the sample proportion according to the number of occurrences A sampling distribution is a graph of a statistic for your sample data. 1. Find the Mean & Standard Deviation. The same five-step procedure is used with either test statistic. In other words, one can take the sampling distribution as the sample mean probability distribution which can attach sample statistics related to a specimen. The sample mean is the average and is calculated as the addition of all the observed outcomes from the sample divided by the total number of events. Step 3: Sum the values from Step 2. Calculate the mean of your data set. An NBA player makes 80% of his free throws (so he misses 20% of them). Only N-1 instead of N changes the calculations. #create empty vector of length n. Specifically, it is the sampling distribution of the mean for a sample size of \(2\) (\(N = 2\)). And the standard deviation of the sampling distribution (σ x ) is determined by the standard deviation of the population (σ), the population size (N), and the sample size (n), as shown in the equation below: σ x = [ σ / sqrt (n) ] * sqrt [ (N - n Jan 1, 2019 · The mean of this sampling distribution is x = μ = 3. But, first, determine the sample standard deviation Aug 30, 2022 · It is calculated as: Sample standard deviation = √Σ (xi – xbar)2 / (n-1) where: Σ: A symbol that means “sum”. There is no lower bound. , the sample proportion) equals approximately 0. 01). To correct for this, instead of taking just one sample from the population, we’ll take lots and lots of samples, and create a sampling distribution of the sample mean. Suppose, the mean of data points in a sample is 90 and the Mar 9, 2019 · Standard deviation is a measure of how much the data in a set varies from the mean. 4\) years. Example 3 Let X - be the mean of a random sample of size 50 drawn from a population with mean 112 and standard deviation 40. Step 3: Find the mean of those squared deviations. 3 = 15 and 50 X (1-0. The standard deviation of the sample is equal to the standard deviation of the population divided by the square root of the sample size. There are two formulas you should use, depending on whether you are calculating the standard deviation based on a sample from a population or based on the whole population. 01) and 0. If the population standard deviation is unknown, calculate the sample standard deviation, s s s. 13. Independent observations within each sample*. σ is the standard deviation of the observations in the sample. Real Life Example. Example. The data follows a normal distribution with a mean score (M) of 1150 and a standard deviation (SD) of 150. Feb 2, 2023 · Find the population standard deviation sigma (𝜎). Solution. Standard deviation formula. S. Subtract the mean from each of the data values and list the differences. Example: Using the empirical rule in a normal distribution You collect SAT scores from students in a new test preparation course. Nov 28, 2020 · Then use the formula to find the standard deviation of the sampling distribution of the sample means: Where σ is the standard deviation of the population, and n is the number of data points in each sampling. Jan 18, 2024 · This normal probability calculator for sampling distributions finds the probability that your sample mean lies within a specific range. Step 2: Subtract the mean from each observation and calculate the square in each instance. First, a large sample ensures that the sampling distribution of \ (\bar {x}\) is nearly normal. 5 hours. ¯. For instance, usually, the population mean estimated value is the sample mean, in a sample space. for(i in 1:n){. Construct a \(90\%\) confidence interval for the mean GPA of all students at the university. 58. Consider a group of 20 people. 71\) with sample standard deviation \(0. You can calculate the p-hat by dividing the sample size by the number of successful outcomes. 6 + 2 (0. The length of time, in hours, it takes a group of people, 40 years old and older, to play one soccer match is normally distributed with a mean of 2 hours and a standard deviation of 0. 5 standard deviations above the mean of the sums. Thirty people from a population of 300 were asked how much they had in savings. Answer Jan 8, 2024 · The Standard Deviation Rule applies: the probability is approximately 0. Sample mean is represented by the symbol. N = Number of observations in population. The type of data available determines which formula to use Apr 30, 2024 · Examples on Sampling Distribution Example 1: Mean and standard deviation of the tax value of all vehicles registered in a certain state are μ=$13,525 and σ=$4,180. The population standard deviation is used if it is known, otherwise the sample standard deviation is used. n: The sample size. x = age that American females first have intercourse. Suppose random samples of size 100 are drawn from the population of vehicles. The sample size is 100, with a mean weight of 65 kg and a standard deviation of 20 kg. It calculates the normal distribution probability with the sample size (n), a mean values range (defined by X₁ and X₂), the population mean (μ), and the standard deviation (σ). The formula for the sample May 30, 2022 · Here are the key differences between the two: Standard deviation: Quantifies the variability of values in a dataset. in Jan 8, 2024 · The central limit theorem states: Theorem 6. Doing so, of course, doesn't change the value of W: W = ∑ i = 1 n ( ( X i − X ¯) + ( X ¯ − μ) σ) 2. x̄ is the mean of the observations in the sample. Solution: Find the mean of the data: First, this section discusses the mean and variance of the sampling distribution of the mean. D. She has a Ph. We can use our Z table and standardize just as we are already familiar with, or can use your technology of choice. n = sample size. Sep 12, 2021 · The Sampling Distribution of the Sample Proportion. Work out the Mean (the simple average of the numbers) 2. 1: One-Sample Means with the t Distribution. Its use it to let us talk about the probability of the sample mean being in a given interval, better understanding the population mean, and so forth. And now of course, the units are back to grams, which makes sense. A light bulb manufacturer claims that a certain type of bulb they make has a mean lifetime of 1000 hours and a standard deviation of 20 hours. 4. And . σ. x – M = 1380 − 1150 = 230. Jan 17, 2023 · Each row represents a sample of size 20 in which each value comes from a normal distribution with a mean of 5. 1 6. Solution (using degrees of freedom = n – 1 = 29) and t α/2 = 2. The mean of the sample mean is \ (\mu_ {\mathrm {\overline {x}}}=\mu=17. 2 μ x ¯ = 8. 7% of values are within 3 standard deviations from the mean. May 25, 2023 · Example 3: The mean and standard deviation of a certain population are and. Sep 17, 2020 · Around 99. Before learning the sample standard deviation formula, let us see when do we use it. Notice the relationship between the mean and standard deviation: The mean is used in the formula to calculate the standard deviation. Check for the needed sample conditions so that the sampling distribution of its proportion p ̂ is normal: The data must be independent. Step 1: Note the number of measurements (n) and determine the sample mean (μ). 3) = 35. This distribution will approach normality as n n X-, the mean of the measurements in a sample of size n; the distribution of X-is its sampling distribution, with mean μ X-= μ and standard deviation σ X-= σ / n. Apr 25, 2024 · If there are 25 successful outcomes in 60 trials, then p-hat (i. 4. In Mathematical terms, sample mean formula is given as: \[\overline{x} = \frac{1}{n} \sum\limits_{i=1}^{n} x \] Dec 11, 2020 · The standard error of the mean indicates how different the population mean is likely to be from a sample mean. Have you ever noticed in class that most students get Cs while a few get As or Fs? The standard deviation of the difference is: σ p ^ 1 − p ^ 2 = p 1 ( 1 − p 1) n 1 + p 2 ( 1 − p 2) n 2. This unit covers how sample proportions and sample means behave in repeated samples. Nov 24, 2020 · Each row represents a sample of size 20 in which each value comes from a normal distribution with a mean of 5. Here are the key takeaways from these two examples: The sampling distribution of a sample mean is approximately normal if the sample size is large enough, even if the population distribution is not normal Each segment (colored in dark blue to light blue) represents one standard deviation away from the mean. Suppose that each package represents an. Step 5: Take the square root. Suppose you're given the data set 1, 2, 2, 4, 6. Mar 26, 2023 · To recognize that the sample proportion \(\hat{p}\) is a random variable. Construct a 95% confidence interval estimate for the population mean. What are the mean μ x̄ and standard deviation σ x̄ of the sample mean x̄? Solution: Kathryn Boddie. 3 hours. Note: For this standard deviation formula to be accurate, our sample size needs to be 10 % or less of the population so we can assume independence. Mean absolute value of the deviation from the mean. State the random variable. = 8. Step 2: Determine how much each measurement varies from the mean. Suppose random samples of size n are drawn from a About this unit. Each package sold contains 4 of these bulbs. ¯x = 8. As explained above in the section on sampling distributions, the standard deviation of a sampling distribution depends on the number of samples. The mean of the sampling distribution (μ x ) is equal to the mean of the population (μ). Let’s take an example to understand z-score calculation better. The central limit theorem says that the sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough. 2. These relationships are not coincidences, but are illustrations of the following formulas. Then I can do it again. The sample proportion p ̂ = 15/50 = 0. Example 2: An unknown distribution has a mean of 80 and a standard deviation of 24. square root of the sample size, in other words: σx̅ =. The mean of the data is (1+2+2+4+6)/5 = 15/5 = 3. Regardless of whether the population has a normal, Poisson, binomial, or any other distribution, the sampling In statistics, a sampling distribution or finite-sample distribution is the probability distribution of a given random-sample-based statistic. Hence the mean is 15 and the standard deviation of the sample mean is 0. There is roughly a 95% chance that p-hat falls in the interval (0. z = 230 ÷ 150 = 1. If random samples of size three are drawn without replacement from the population consisting of four numbers 4, 5, 5, 7. Q3. 58, 0. Compare your calculations with the population parameters. Well now, when I calculate the sample mean, the average of one and three or the mean of one and three is going to be equal to two. (where n 1 and n 2 are the sizes of each sample). In a practical situation, when the population size N is large it becomes difficult to obtain value x i for every observation in the population and hence it becomes difficult to calculate the standard deviation (or variance) for the population. It is algebraically simpler, though in practice less robust, than the average absolute deviation. From the central limit theorem, we know that as n gets larger and larger, the sample means follow a normal distribution. s = sample standard deviation. A normal distribution curve can represent hundreds of situations in real life. What are the mean and standard deviation of the sample mean? Solution 3: Since we can say that =. Apr 22, 2024 · Therefore, the sample standard deviation is 1. Example #2. The standard deviation of the sample means is σ¯. Then work out the mean of those squared differences. A sample of size 80 is drawn randomly from the population. To learn what the sampling distribution of \(\hat{p}\) is when the sample size is large. If 36 samples are randomly drawn from this population then using the central limit theorem find the value that is two sample deviations above the expected value. Central limit theorem is applicable for a sufficiently large sample sizes (n ≥ 30). For example, the standard deviation for a binomial distribution can be computed using the formula. Apr 7, 2020 · A sampling distribution is a probability distribution of a certain statistic based on many random samples from a single population. Jul 6, 2022 · The distribution of the sample means is an example of a sampling distribution. Oct 10, 2022 · The distribution of the sample means is an example of a. Sep 12, 2021 · There are two formulas for the test statistic in testing hypotheses about a population mean with small samples. Answer: The sample mean of 60, 57, 109, 50 is 69. Keep reading to learn more Jun 7, 2024 · A Worked Example. Sampling distributions are crucial because they place the value of your sample statistic into the broader context of many other possible values. And for this sample of two, it's going to be 1. Divide the population standard deviation—or sample standard deviation—by the square root of the sample size. (The subscript 4 is there just to remind us that the sample mean is based on a sample of size 4. Our data set has 8 values. n * (1 - p) ≥ 10. A sample of size n = 50 is drawn randomly from the population. The z score for a value of 1380 is 1. SRS. Brian’s research indicates that the cheese he uses per pizza has a mean weight of 7. Write the probability The standard deviation of the sample mean X−− that we have just computed is the standard deviation of the population divided by the square root of the sample size: 10−−√ = 20−−√ / 2–√. Find the sum that is 1. Evaluate the standard deviation. V a r ( X ¯) = σ 2 n. Standard error: Quantifies the variability between samples drawn from the same population. Calculation. The data are randomly sampled from a population so this condition is true. 95 that p-hat falls within 2 standard deviations of the mean, that is, between 0. Now, we can take W and do the trick of adding 0 to each term in the summation. Unbiased estimate of variance. The np ̂≥10 and n (1-p ̂)≥10. For large samples, the sample proportion is approximately normally distributed, with mean μP^ = p μ P ^ = p and standard deviation σP^ = pq n−−√ σ P ^ = p q n. 7% of scores are within 3 standard deviations of the mean. For the test of one group mean we will be using a t test statistic: Test Statistic: One Group Mean. We will see in Section 5. Regardless of whether the population has a normal, Poisson, binomial, or any other distribution, the sampling This Normal distribution is the distribution of the sample mean. Example: Standard deviation in a normal distribution You administer a memory recall test to a group of students. Step 4: Divide by the number of data points. 2) The standard deviation of x̅ equals the population standard deviation divided by the. It assesses how far a sample statistic likely falls from a population parameter. This standard deviation formula is exactly correct as long as we have: Independent observations between the two samples. So the standard deviation of the sampling distribution for the difference in sample means over here is going to be the square root of 5/8. In the examples so far, we were given the population and sampled from that population. Following the empirical rule: Also Check: Difference Between Variance and Standard Deviation. The mean is now x (called "x-bar") for sample mean, instead of μ for the population mean, And the answer is s (for sample standard deviation) instead of σ. To find the mean and standard deviation of this sampling distribution of sample means, we can first find the mean of each sample by typing the following formula in May 31, 2019 · Consider the fact though that pulling one sample from a population could produce a statistic that isn’t a good estimator of the corresponding population parameter. 96 oz, with a standard deviation of . e. x = 1380. The data follows a normal distribution with a mean score of 50 and a standard deviation of 10. seed(0) #define number of samples. Step 3: Square all the deviations determined in step 2 and add altogether: Σ (x i – μ)². Answer a. Standard deviation formula is given by the root of summation of square of the distance to the mean divided by number of data points. Sample Standard Deviation Formula. 13 σ x ¯ = σ n = 1 60 = 0. To understand the meaning of the formulas for the mean and standard deviation of the sample proportion. where p is the probability of success, q = 1 - p, and n is the number of elements in the sample. sample_means = rep(NA, n) #fill empty vector with means. 3 and a standard deviation of 9. To find the standard deviation from a sample, the sample standard deviation formula applies, which is: Oct 8, 2018 · In this situation, the mean will vary from sample to sample and form a distribution of sample means. Oct 23, 2020 · Around 99. Question A (Part 2) To calculate the standard deviation of those numbers: 1. x i = ith observation in the population. The following code shows how to generate a sampling distribution in R: set. For this simple example, the distribution of pool balls and the sampling distribution are both discrete distributions. xi: The ith value in the sample. Question: During a survey, 6 students were asked how many hours per day they study on an average? Their answers were as follows: 2, 6, 5, 3, 2, 3. Use the below-given data for the calculation of the sampling distribution. 41 The higher the standard deviation, the more spread out the values are from the mean, while a lower standard deviation indicates that the values tend to be closer to the mean. The Jan 21, 2021 · Example \(\PageIndex{1}\) Finding the Probability Distribution, Mean, Variance, and Standard Deviation of a Binomial Distribution. 3) If x is normally distributed, so is x̅, regardless of sample size. 3. The sampling distribution of a sample proportion p ^ has: μ p ^ = p σ p ^ = p ( 1 − p) n. And this is approximately going to be equal to, get my calculator out, 5 divided by 8 equals, and then we take the square root of that, and Here's a quick preview of the steps we're about to follow: Step 1: Find the mean. b. Solution: To find: Sample mean Sum of terms = 60 + 57 + 109 + 50 = 276 Number of terms = 4 Using sample mean formula, mean = (sum of terms)/ (number of terms) mean = 276/4 = 69. Then, it talks about the properties of the sampling distribution for differences between means by giving the formulas of both mean and variance The Central Limit Theorem helps us to describe the distribution of sample means by identifying the basic characteristics of the samples - shape, central tendency and variability. 5. 53. For example, 2σ means two standard deviations from the mean. Work through each of the steps to find the standard deviation. To find the mean and standard deviation of this sampling distribution of sample means, we can first find the mean of each sample by typing the following formula in Oct 9, 2020 · Step 2: Divide the sum by the number of values. 0452 for a 95% confidence level): The key takeaways from this lesson are summarized below. A sampling distribution shows every possible result a statistic can take in every possible sample from a population and how often each result happens - and can help us use samples to make predictions about the chance tht something will occur. It assesses how far a data point likely falls from the mean. It also shows how central limit theorem can help to approximate the corresponding sampling distributions. 55. The variance of the sampling distribution of the mean is computed as follows: \[ \sigma_M^2 = \dfrac{\sigma^2}{N}\] That is, the variance of the sampling distribution of the mean is the population variance divided by \(N\), the sample size (the number of scores used to compute a mean). 8 hours and 2. 62) for samples of this size. Mar 11, 2023 · Z-scores assuming the sampling distribution of the test statistic (mean in most cases) is normal and transform the sampling distribution into a standard normal distribution. in Applied Mathematics from the University of Wisconsin-Milwaukee, an M. 01 oz. Range. The motivation in Chapter 4 for requiring a large sample was two-fold. The probability of success outcome. But they do not affect the calculations. Mar 14, 2024 · Take the example of the female population. M = 1150. SD = 150. While, technically, you could choose any statistic to paint a picture, some common ones you’ll come across are: Mean. It is the average of all the measurements. • The estimated standard deviation of the sample proportion is σbbp= s pb(1−pb) n r 1− n N. Jan 8, 2024 · For example, if sigma (σ) is the population’s standard deviation, then the Mean for all the Means (x-bars or X̅ ) will exactly represent the population mean mu (μ). = 400 8 = 50. This page titled 8. (4) • The Central Limit Theorem applies directly, because we are really Nov 23, 2020 · Generate a Sampling Distribution in R. So if we choose our sample size large enough and ensure that our sample is randomly selected we can state the the sample mean that we calculate is within some range of the actual population mean (based on our sample standard deviation) with a certain degree of certainty (usually 95% or 99. Step 2: Divide the difference by the standard deviation. Assume that the numerical population of GPAs from which the sample is taken has a normal distribution. The sample mean (x̄) was $1,500, with a sample standard deviation of $89. When population sizes are large relative to sample sizes, the standard deviation of the difference between sample proportions (σ d) is approximately equal to: σ d = sqrt { [P 1 (1 - P 1) / n 1] + [P 2 (1 - P 2) / n 2] } It is straightforward to derive this equation, based on material covered in . As you can see, we added 0 by adding and subtracting the sample mean to the quantity in the numerator. ) And, the variance of the sample mean of the second sample is: V a r ( Y ¯ 8 = 16 2 8 = 32. Simply enter the appropriate values for a given Then, for samples of size n, 1) The mean of x̅ equals the population mean, , in other words: μx̅ = μ. 1 that if the population data are nearly normal, then \ (\bar {x}\) is also nearly normal regardless of the. For samples of a single size n n, drawn from a population with a given mean μ μ and variance σ2 σ 2, the sampling distribution of sample means will have a mean μX¯¯¯¯¯ = μ μ X ¯ = μ and variance σ2X = σ2 n σ X 2 = σ 2 n. Calculate the mean and standard deviation of this sampling distribution. The formula for central limit theorem can be stated as follows: \ [\LARGE \mu _ {\overline {x}}=\mu\] \ (\begin {array} {l Apr 23, 2022 · The distribution shown in Figure \(\PageIndex{2}\) is called the sampling distribution of the mean. And let's say I get a one and I get a three. Summary. Find the probability that the sum of the 80 values (or the total of the 80 values) is more than 7,500. Apr 2, 2023 · An unknown distribution has a mean of 90 and a standard deviation of 15. The variance of this sampling distribution is s 2 = σ 2 / n = 6 / 30 = 0. sampling distribution. 2 . of bulbs, and we calculate the sample mean lifetime x ¯ of the bulbs in each package. Note that structure of this formula is similar to the general formula for a test statistic: s a m p l e s t a t i s Remeber, The mean is the mean of one sample and μX is the average, or center, of both X (The original distribution) and . Then for each number: subtract the Mean and square the result. The standard deviation of a random variable, sample, statistical population, data set, or probability distribution is the square root of its variance. 6 – 2 (0. Subtract 3 from each of the values 1, 2, 2, 4, 6. Example Question based on Standard Deviation Formula. Statisticians refer to this type of distribution as a sampling distribution. A sample is large if the interval [p − 3σp^, p + 3σp^] [ p − 3 σ p ^, p + 3 σ p ^] lies wholly within the interval W = ∑ i = 1 n ( X i − μ σ) 2. Apr 19, 2023 · It is denoted using z and calculated as: Z = (x-x̄)/σ. n * p ≥ 10, where p is the sample proportion. Example 2: Five friends having heights of 110 units, 115 units, 109 units, 112 units, and 114 units respectively. The mean of the sampling distribution is very close to the population mean. (b) What is the probability that sample proportion p-hat Apr 23, 2022 · 5. Step 2: For each data point, find the square of its distance to the mean. 51\). 7%). (Remember that the standard deviation for X ¯ X ¯ is σ n σ n. If we get an astronomically large sample size, the standard deviation will be astronomically small. n = 10000. The larger the value of standard deviation, the more the data in the set varies from the mean. Find the standard deviation given that he shoots 10 free throws in a game. This statistics video tutorial explains how to use the standard deviation formula to calculate the population standard deviation. Solution: We know that mean of the sample equals the mean of the population. Following the empirical rule: Around 68% of scores are between 40 and 60. But, if we pick another sample from the same population, it may give a different value. Let us take the sample standard deviation formula example of an office in New York where around 5,000 people work, and a survey has been carried out on a sample of 10 people to determine the average age of the working population. In the formula, n is the number of values in your data set. Step 1: Subtract the mean from the x value. The sampling distribution for a sample proportion will be normally distributed when: Population size (N) is at least 10 times sample size (n). Even though the original random variable is not normally distributed, the sample size is over 30, by the central limit theorem the sample mean will be normally distributed. Suppose random samples of size 100 are drawn from the population. Formula. Take the square root of that and we are done! • The sample mean is x = pb, the sample proportion • The population standard deviation is σ = q p(1−p), but we never know it, so we use the estimate q pb(1−pb). μ 0 = hypothesized population mean. Part 2: Find the mean and standard deviation of the sampling distribution. = 400. 2. where, x is an observation in the sample. An unknown distribution has a mean of 90 and a standard deviation of 15. The larger n gets, the smaller the standard deviation gets. ¯x = σ √n = 1 √60 = 0. Solution: Because the sample size of 60 is greater than 30, the distribution of the sample means also follows a normal distribution. 50 X 0. There are two commonly used forms of the standard deviation formula: one for a population and one for a sample. Standard deviation of the sample. The standard deviation of the sampling distribution is smaller than the standard deviation of the population. √n. The mean tells us that in our sample, participants spent an average of 50 USD on their restaurant bill. Apr 23, 2022 · Sampling Variance. In this case, it would be the sample mean which is used to estimate the population mean. Find the probability that the sample mean is between 1. One test statistic follows the standard normal distribution, the other Student’s \(t\)-distribution. xbar: The mean of the sample. Standard Deviation of Sampling Distribution. x ― = sample mean. μ = Population mean. The smaller the value of standard deviation, the less the data in the set varies from the mean. Nov 5, 2020 · The z score tells you how many standard deviations away 1380 is from the mean. So the distribution of sample means helps us to find the probability associated with each specific sample. σ = √ (∑ (xi – μ) 2 /N) Here, σ = Population standard deviation. Help the researcher determine the mean and standard deviation of the sample size of 100 females. 0 license and was authored, remixed, and/or curated by Anonymous via source Sample question: If a random sample of size 19 is drawn from a population distribution with standard deviation α = 20 then what will be the variance of the sampling distribution of the sample mean? Step 1: Figure out the population variance . If an arbitrarily large number of samples, each involving multiple observations (data points), were separately used in order to compute one value of a statistic (such as, for example, the sample mean or sample variance) for each sample, then the sampling The procedure to calculate the standard deviation is given below: Step 1: Compute the mean for the given data set. Find the sample mean $$\bar X$$ for each sample and make a sampling distribution of $$\bar X$$. When looking at a person’s eye color, it turns out that 1% of people in the world has green eyes ("What percentage of," 2013). Therefore, the variance of the sample mean of the first sample is: V a r ( X ¯ 4) = 16 2 4 = 64. Mar 26, 2023 · The population standard deviation is used if it is known, otherwise the sample standard deviation is used. Calculate the square root of your sample size. 2: Large Sample Tests for a Population Mean is shared under a CC BY-NC-SA 3. ) This means that the sample mean x ¯ x ¯ must be close to the population mean μ. ob je zo vx nu op mm pk no iv