What measure of central tendency where it is the middle value of the observation?
There are various methods to measure the central value of an observation. For instance, average, mean, median, mode, etc. Show
AverageThe average value of a characteristic is the one central value around which all other observations are distributed. In any large series, nearly 50% of observations lie above the central value whereas the other 50% lie below the central value. It indicates how the values lie close to the center. **Average is the measure that indicates the central tendency or concentration of all other observations around the central value. There are three types of averages or measures of central position or central tendency -mean, median, and mode. Get Free Netflix Now Best safe and secure cloud storage with password protection GPL Themes For Free Get Envato Elements, Prime Video, Hotstar and Netflix For Free Best Money Earning Website 100$ Day Best ever Chat Forum MeanThis is an arithmetic average of a collection of values. It is the sum of observations divided by the number of observations. Let,the observations are x1,x2,x3,…. Mean=∑x/n (n=number of observations) Source: wallstreetmojoMean in case of continuous series:
x¯=∑fm/∑f=352/21=16.76 MedianWhen all the observations of a variable one arranged in either ascending or descending order, the middle observation is known as the median. The median is a locational average & it is that value that divides the series into two equal shares in a series of observations. The median equally divides the number of items of observations. Source: WikipediaWith an even number of observationsThe Median will be the average of two values according to the following formula. when,n=10 n/2=10/2=5th (n+2)/2=(10+2)/2=6th, Here median will be the average of 5th & 6th value of the series. Example:4,6,7,8,8,9,11,12,12,14 Here,5th value=8 & 6th value=9 Now,median=(8+9)/2=17/2=8.5 With odd number of observationsNumber of nodules/root:7,6,9,8,9,10,6,7,10 Arranged in ascending order:6,6,7,7,8,9,9,10,10 Median=(n+1)/2=(9+1)/2=5th value=8 Median in case of classified data
Median=L+{(n/2-Fc)/fm}×h=48+{(70/2-31)/25}×6=48.96 L= lower value of the highly frequent class (48-53) n=number of observations [n=70],[2f=70] Fc=cumulative value of the previous class of the median class [Fc=31] fm= frequency of median class h=class interval[h=6] ModeThis is the most frequently occurring observation in a series. Mode is the type of average that refers to the most common or most regularly occurring value in a series of data. Identifying Mode. Source: minakshi maths academyCalculation of mode from classified data
Mode=L+{f1/(f1+f2)}×h=61+{4/(4+3)}×10=61+0.57×9=66.71 Here, L=lower value of the modal class=61 f1=frequency of the modal class – frequency of the previous class=12-8=4 f2=frequency of the modal class – frequency of the next class=12-9=3 h=class interval=10 **Out of three measures of central tendency mean is better and utilized more often because it uses all the observations in the data and is further used in the tests of significance Symmetrical DistributionSymmetrical distribution is a distribution in which the values of variables occur at regular frequencies, and the mean, median, and mode occur at the same point. Unlike asymmetrical distribution, symmetrical distribution does not skew. Source: passyourmath**In short, the distributions in which both sides are the same(mirror image) about the middle ordinate is called symmetrical distribution. Asymmetrical DistributionAsymmetrical distribution is a distribution in which the values of variables occur at irregular frequencies and the mean, median, and mode occur at different points. An asymmetric distribution exhibits skewness. Source: researchgateSkewed DistributionSome distributions are steep on one side and have a long tail on the other side. This characteristic of a distribution is called skewness. If the tail is on the right side the distribution is said to be skewed right. This type of skewness is called positive skewness. Source: chegg**Biological phenomena are very commonly skewed right. Source: towardsdatascienceIf the tail is on the left side, the distribution is said to be skewed left. This type of skewness is called negative skewness. **Skewness is a measure of symmetry, or to say more certainly, the lack of symmetry. KurtosisSimilar to skewness, kurtosis is a statistical measure that is used to describe the distribution. Kurtosis is a measure of whether or not the data are heavy-tailed or light-tailed relative to a normal distribution. That is, data sets with high kurtosis tend to own heavy tails or outliers. Data sets with low kurtosis tend to own light tails or lack of outliers. Source: researchgate**Kurtosis=3 depicts the normal distribution OutlierIn statistics, an outlier is a data point that differs considerably from other observations. An outlier may be due to variability in the measurement or it may suggest experimental error; the latter are generally excluded from the data set. An outlier can cause serious issues in statistical analyses. Q&A1. Which of the following data sets has a median of 3? (A) 3, 3, 3, 3, 3 (B) 2, 5, 3, 1, 1 (C) 1, 2, 3, 4, 5 (D) 1, 2, 4, 4, 4 (E) Choices (A) and (C) 2. To the nearest tenth, what is the mean of the following data set? 14, 14, 15, 16, 28, 28, 32, 35, 37, 38 3. To the nearest tenth, what is the mean of the following data set? 15, 25, 35, 45, 50, 60, 70, 72, 100 4. To the nearest tenth, what is the mean of the following data set? 0.8, 1.8, 2.3, 4.5, 4.8, 16.1, 22.3 5. To the nearest thousandth, what is the mean of the following data set? 0.003, 0.045, 0.58, 0.687, 1.25, 10.38, 11.252, 12.001 6. To the nearest tenth, what is the median of the following data set? 6, 12, 22, 18, 16, 4, 20, 5, 15 7. To the nearest tenth, what is the median of the following data set? 18, 21, 17, 18, 16, 15.5, 12, 17, 10, 21, 17 8. To the nearest tenth, what is the median of the following data set? 14, 2, 21, 7, 30, 10, 1, 15, 6, 8 9. To the nearest hundredth, what is the median of the following data set? 25.2, 0.25, 8.2, 1.22, 0.001, 0.1, 6.85, 13.2 10. Compare the mean and median of a data set that has a distribution that is skewed right. 11. Compare the mean and median of a data set that has a distribution that is skewed left. 12. Compare the mean and the median of a data set that has a symmetrical distribution. 13. Which measure of center is most resistant to (or least affected by) outliers? 1. Ans) E. Choices (A) and (C) (3, 3, 3, 3, 3; 1, 2,
3, 4, 5) 2. Ans) 25.7 Use the formula, ¯x=∑x/n where ̄x is the mean, ∑ represents the sum of the data values, and n is the number of values in the data set. In this case, x = 14 + 14 + 15 + 16 + 28 + 28 + 32 + 35 + 37 + 38 = 257, and n = 10. So the mean is 257 10 =25. 3. Ans) 52.4 ¯x=∑x/n where ̄x is the mean, ∑ represents the sum of the data values, and n is the number of values in the data set. In this case, x = 15 + 25 + 35 + 45 + 50 + 60 + 70 + 72 + 100 = 472, and n = 9. So the mean is 472 9 =52.4444 4. Ans) 7.5 ¯x=∑x/n where
̄x is the mean, ∑ represents the sum of the data values, and n is the number of values in the data set. In this case, x = 0.8 + 1.8 + 2.3 + 4.5 + 4.8 + 16.1 + 22.3 = 52.6, and n = 7. So the mean is 52.6 7 =7.5143 5. Ans) 4.525 ¯x=∑x/n where ̄x is the mean, ∑ represents the sum of the data values, and n is the number of values in the data set. In this case, x = 0.003 +
0.045 + 0.58 + 0.687 + 1.25 + 10.38 + 11.252 + 12.001 = 36.198, and n = 8. So the mean is 36.198 8 =4.52475 6. Ans) 15.0 7. Ans) 17.0 8. Ans) 9.0 9. Ans) 4.04 10. Ans) The mean will have a higher value than the median. 11. Ans) The mean will have a lower value than the median. 12. Ans) The mean and median will be fairly close together. 13. Ans) median Reference Statistics: 1,001 Practice Problems For Dummies Which of the measure of central tendency that is considered as the middle value in a list from smallest to largest of the data array?The median is the middle value. It is the value that splits the dataset in half, making it a natural measure of central tendency. To find the median, order your data from smallest to largest, and then find the data point that has an equal number of values above it and below it.
Which measures of central tendency is when the score in the middle of the set of items cuts or divides the set into two groups?the median, symbolized Mdn, is the middle score. It cuts the distribution in half, so that there are the same number of scores above the median as there are below the median. Because it is the middle score, the median is the 50th percentile. Here's an example.
Which measure of central tendency refers to the middle value that splits the dataset into half?The median is the middlemost number. In other words, it's the number that divides the distribution exactly in half such that half the cases are above the median, and half are below. It's also known as the 50th percentile, and it can be calculated for ordinal and interval/ratio variables.
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