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• 1. Project Submitted On“Business Statistics” By Rajkumar Jangid (BBA 1st Year) 106/10 civil lines,Ajmer 305001 Website: www.dezyneecole.com 1
• 2. ARITHMATIC MEAN The mean is obtained by dividing the some of observed valuesby the number of observation, n. Although data points fall aboveor on the mean it can be considered a good estimate for productingsubsequent data points.X = X N Calculation of Mean 1. Individual Series : In individual series we can find out arithmetic mean by two methods : a. Direct Method : In direct method we find out the total of all values and divided by total of items. b. Shortcut Method : It is used on account of the fact that at it makes simple calculation. 2. Discrete Series : In discrete series also arithmetic mean is calculated according two methods : a. Direct Method : Under direct method arithmetic mean is calculated in the following way : i. Every size is multiplied with its frequency. ii. Some of multiplied values is found out. iii. This some is divided by total number of frequency. b. Shortcut Method : This method involves the following procedure : i. Take a value as assumed mean. ii. Find the deviations of all the values from assumed mean. 2
• 3. iii. Find the products by multiplying deviations with respective frequency. iv. Find the sum of the products.3. Continuous Series : Arithmetic mean in continuous series may be calculated by the following methods : a. Direct Method : While calculating arithmetic mean by direct method in a continuous series, we convert the series in a discrete from by finding out the mid values and rest of the procedure is the same as that in case of discrete series. b. Shortcut Method : Under this method also we convert the series into discrete from by finding out the mid-values. c. Step Deviation Method : Under this method, the deviations are divided by a common factor (values), so as to make the figures smaller for easier calculation. ADVANTAGES OF ARITHMETIC MEAN  Simplicity : Arithmetic mean is easy to understand and calculate.  Certainty : Arithmetic mean is always also determine is certainty while for some other averages there is no certainty that they will be determined.  Based on all the values : Arithmetic is based on all the values. 3
• 4. DISADVANTAGES OF ARITHMETIC MEAN  Unreal : Sometimes, in arithmetic we find such a value which seems to be unreal  Cannot be Calculated by Inspection Only : The arithmetic mean cannot be calculated by Inspection only while some other averages can be calculated by inspection.  All Real Values must be known : Unless we know all the real values we cannot calculate arithmetic mean. MODE The mode of a set of data is the value which occurs mostfrequently. The excel syntax for the mode is MODE. Calculation of Mode 1. Individual Series : In an individual series we cannot calculate mode without converting it into a discrete series as the frequency of every size is one. 2. Discrete Series : In discrete series we calculated mode by the following methods : a. By Inspection : Where the distribution of frequencies is regular, mode may be calculated by a simple inspection. 4
• 5. b. By Grouping Method : Where the distribution of frequencies is irregular, mode is calculated by grouping, since by inspection the mode can’t be found out.3. Continues Series : While calculating mode in continuous series, we must see that the series is exclusive and class intervals are equal. Formula : 1 Z = l1+ *i 1 + 2 ADVANTAGES OF MODE Calculation Simple : Calculation of mode is simple. Sometimes in individual and discrete series it can be found out by inspection only. Graphical Representation : Mode can be calculated graphically also. Use in Quantitative Facts : Mode is also used for describing the quantitative data. DISADVANTAGES OF MODE Uncertainty : Calculation is mode not always certain Not Based on All the Values : Mode is not based on all the values of a series as mode is not affected by extreme values. 5
• 6.  Algebraic Treatment Not Possible : After calculating mode we cannot algebraically analyse it as it is not based on all the items. MEDIAN The median is the middle value of a set of data containing anodd number of values, or the average of the two middle value of aset of data with an even number of values. The median especiallyhelpful when separating data into two equal size bins. The excelsyntax to find the median is MEDIAN. Calculation of Median 1. Individual Series : For calculating median in the individual series, we will arrange the series in ascending or descending order. Then write the serial numbers. After arrange and write the serial number we will use the following formula : n 1 M= 2 2. Discrete Series : For calculating the median in discrete series, we will arrange the series in ascending or descending order. Then find out cumulative frequencies. Use this formula to out the median. n 1 M= 2 6
• 7. 3. Continuous Series : In continuous series the median will be calculated, convert the series into exclusive continuous form. The find out cumulative frequencies. After calculating it find median group the median group is seen in cumulative frequency. We use this formula : M = l1+ i (m-c) f ADVANTAGES OF MEDIAN Simple : It can be calculated as well as understood easily by all. Specific : It is specific in every type of series, while other averages may not be clear sometimes. Least Affected by Extreme Values : Median is least affected by extreme values. DISADVANTAGE OF MEDIAN Lack of Algebraic Treatment : Algebraic treatment of median is not possible. Effect of Sampling : Median is affected much by sampling as comparised to other averages. Arrangement Required : To calculate median, data are the arrange in an ascending order or a descending order while it is not necessary of other averages. 7
• 8. RELATIONSHIP BETWEEN MEAN, MODE AND MEDIANWith the help of Numerical Question :Ques. (a) If the value of mean ( X ) = 5 and mode (Z) = 5, find outthe value of median (M). (b) In a moderately symmetrical distribution : (i) Median and Mean are 24 and 25, calculate mode. (ii) Mode and Median are 20 and 22, calculating arithmetic mean. (iii) Mode and mean are 30 and 33, calculate median.Solve : (a) given: X =5, Z=5 Z= 3M – 2 X or -3M = 2 X +ZM = 2 X +Z = 2 * 5 + 5 = 10 + 5 = 15 = 5 3 3 3 3(b) (i) Z = 3M - 2 X or Z = 3*24 – 2 * 25 or Z = 72 -50 = 22 Mode is 22 (ii) Z = 3M/0 2 X or 2 X = 3M – Z 2 X = 3 *22 -20 or 66 – 20 = 46 8
• 9. X = 46/2 = 23 Arithmetic mean = 23(iii)Z = 3M - 2 X or -3M = 2 X + Z orM = 2 X +Z = 2*33+30 or 66+30 or 3 3 3 96 = 32 3 Median = 32 9