what is the formula of assumed mean?
Answers
Answer:
Step-by-step explanation:
The method depends on estimating the mean and rounding to an easy value to calculate with. This value is then subtracted from all the sample values. When the samples are classed into equal size ranges a central class is chosen and the count of ranges from that is used in the calculations. For example, for people's heights a value of 1.75m might be used as the assumed mean.
For a data set with assumed mean x0 suppose:
{\displaystyle d_{i}=x_{i}-x_{0}\,} {\displaystyle d_{i}=x_{i}-x_{0}\,}
{\displaystyle A=\sum _{i=1}^{N}d_{i}\,} {\displaystyle A=\sum _{i=1}^{N}d_{i}\,}
{\displaystyle B=\sum _{i=1}^{N}d_{i}^{2}\,} {\displaystyle B=\sum _{i=1}^{N}d_{i}^{2}\,}
{\displaystyle D={\frac {A}{N}}\,} {\displaystyle D={\frac {A}{N}}\,}
Then
{\displaystyle {\overline {x}}=x_{0}+D\,} {\displaystyle {\overline {x}}=x_{0}+D\,}
{\displaystyle \sigma ={\sqrt {\frac {B-ND^{2}}{N}}}\,} {\displaystyle \sigma ={\sqrt {\frac {B-ND^{2}}{N}}}\,}
or for a sample standard deviation using Bessel's correction:
{\displaystyle \sigma ={\sqrt {\frac {B-ND^{2}}{N-1}}}\,} {\displaystyle \sigma ={\sqrt {\frac {B-ND^{2}}{N-1}}}\,}
Example using class ranges
Where there are a large number of samples a quick reasonable estimate of the mean and standard deviation can be got by grouping the samples into classes using equal size ranges. This introduces a quantization error but is normally accurate enough for most purposes if 10 or more classes are used.
For instance with the sample:
167.8 175.4 176.1 166 174.7 170.2 178.9 180.4 174.6 174.5 182.4 173.4 167.4 170.7 180.6 169.6 176.2 176.3 175.1 178.7 167.2 180.2 180.3 164.7 167.9 179.6 164.9 173.2 180.3 168 175.5 172.9 182.2 166.7 172.4 181.9 175.9 176.8 179.6 166 171.5 180.6 175.5 173.2 178.8 168.3 170.3 174.2 168 172.6 163.3 172.5 163.4 165.9 178.2 174.6 174.3 170.5 169.7 176.2 175.1 177 173.5 173.6 174.3 174.4 171.1 173.3 164.6 173 177.9 166.5 159.6 170.5 174.7 182 172.7 175.9 171.5 167.1 176.9 181.7 170.7 177.5 170.9 178.1 174.3 173.3 169.2 178.2 179.4 187.6 186.4 178.1 174 177.1 163.3 178.1 179.1 175.6
The minimum and maximum are 159.6 and 187.6 we can group them as follows rounding the numbers down. The class size (CS) is 3. The assumed mean is the centre of the range from 174 to 177 which is 175.5. The differences are counted in classes.
Observed numbers in ranges
Range tally-count frequency class diff freq×diff freq×diff2
159—161 / 1 −5 −5 25
162—164 //// / 6 −4 −24 96
165—167 //// //// 10 −3 −30 90
168—170 //// //// /// 13 −2 −26 52
171—173 //// //// //// / 16 −1 −16 16
174—176 //// //// //// //// //// 25 0 0 0
177—179 //// //// //// / 16 1 16 16
180—182 //// //// / 11 2 22 44
183—185 0 3 0 0
186—188 // 2 4 8 32
Sum N = 100 A = −55 B = 371
The mean is then estimated to be
{\displaystyle x_{0}+CS\times {\frac {A}{N}}=175.5+3\times -55/100=173.85} {\displaystyle x_{0}+CS\times {\frac {A}{N}}=175.5+3\times -55/100=173.85}
which is very close to the actual mean of 173.846.
The standard deviation is estimated as
{\displaystyle CS{\sqrt {\frac {B-{\frac {A^{2}}{N}}}{N-1}}}=5.57} {\displaystyle CS{\sqrt {\frac {B-{\frac {A^{2}}{N}}}{N-1}}}=5.57}
a+{sigma fidi}
—
sigma fi
fidi means product of frequency and di which is obtained by doing Xi - A.