UPSKILL MATH PLUS

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Learn moreLet us consider the steps for finding the mean of grouped data using the step deviation method.

**Steps**:

1. Calculate the midpoint of the class interval and name it as \(x_i\).

2. From the data of \(x_i\), choose any value(preferably in the middle) as the assumed mean(\(a\)).

3. Determine the deviation (\(d = x_i - a\)) for each class.

4. Determine the deviation (\(u = \frac{x_i - a}{h}\) where \(h\) is the class size) for each class.

5. Multiply the frequency and \(u_i\) of each class interval and name it as \(f_iu_i\).

6. Calculate the mean by applying the formula \(\overline X = a + \left[\frac{\sum fd}{\sum f} \times h \right]\).

Example:

Find the mean of the following frequency distribution:

Class interval | \(30 - 40\) | \(40 - 50\) | \(50 - 60\) | \(60 - 70\) | \(70 - 80\) |

Frequency | \(124\) | \(156\) | \(200\) | \(10\) | \(10\) |

**Solution**:

Let the assumed mean be \(a = 55\) and class width \(h = 10\).

Class interval | Frequency\(f_i\) | Midpoint\(x_i\) | Deviation \(d_i = x_i - 55\) | \(u_i = \frac{x - a}{h}\) | \(f_iu_i\) |

\(30 - 40\) | \(124\) | \(35\) | \(-20\) | \(-2\) | \(-248\) |

\(40 - 50\) | \(156\) | \(45\) | \(-10\) | \(-1\) | \(-156\) |

\(50 - 60\) | \(200\) | \(55\) | \(0\) | \(0\) | \(0\) |

\(60 - 70\) | \(10\) | \(65\) | \(10\) | \(1\) | \(10\) |

\(70 - 80\) | \(10\) | \(75\) | \(20\) | \(2\) | \(20\) |

Total | \(\sum f_i = 500\) | \(\sum f_iu_i = -374\) |

We know that the mean of the grouped frequency distribution using the step deviation method can be determined using the formula, \(\overline X = a + \left[\frac{\sum f_iu_i}{\sum f_i} \times h \right]\).

Substituting the known values in the above formula, we have:

\(\overline X = 55 + \left[\frac{-374}{500} \times 10 \right]\)

\(\overline X = 55 - 7.48\)

\(\overline X = 47.52\)

Therefore, the mean of the given data is \(47.52\).